Lance Wabisuke-Hamilton from One Piece: Parallel Works has a mysterious illness that causes him to fall asleep at random times and makes him physically weak. However, Word of God states that Lance's condition is not life-threatening.
In the Avatar: The Last Airbender fanfic Three Years At Sea, Zuko was "sickly" for most of his childhood, making him appear even weaker in comparison to Azula in his father's eyes.
Tim in His Own Personal Heaven is diagnosed with colitis and fell out of touch with all his friends. He only finds consolation in fantasizing about himself with six colorful mares.
Twitchy in the Homestuck fanfiction series Golden Age frequently has seizures and appears to have brain-damage. This is because the Alternian Empire planted a psychic centipede-type thing in his head to track and disable the United Galaxies ship.
In Live a Hero (MHA), Izuku's years of malnutrition and inability to get a good night's sleep has left his immune system weakened, causing him to frequently break into fevers.
The Job Gone Wrong makes use of Gambit's canon intolerance to medication, with him having a bad reaction to something he was drugged with at a government facility.
In Which Guzma Is Frustrated: Acerola's father died of an illness that left him bedbound, weak, and with a bad cough.
Towards the Sun: Zuko's poor health threaten his life and his reign as Fire Lord. Zuko ends up developing heart arrhythmia due to being constantly attacked with lightning by Azula. Not helping him was his poor conditions in his Tailor-Made Prison where he lacked sunlight, movement and just enough food and water for him not to die. Add his poor sleeping habits and lack of rest makes a perfect storm of poor health that leads to Zuko suffering a thankfully non-fatal heart attack. Iroh was able to use Zuko's poor health to overthrown him as Fire Lord.
The Arrangement: Link is a sickly child. It is thought he died after being kidnapped, however he was really hidden away and raised amongst the Gerudo. As Link grows, he becomes healthier.
The Yu-Gi-Oh! ARC-V story (æ)mæth has the original character Fuyu, a Duelist who suffered from a combination of diseases that left him bedridden in his childhood. After meeting Hokuto, he gets better, but he must still wear special clothing to protect his pale skin from the sun, and he often practices Dueling in a micro-gravity Action Field that turns his frail physique into an advantage.
The Scum Villain's Self-Saving System: Ren Zha Fanpai Zijiu Xitong fanon tends to portray Shen Yuan as suffering from chronic illness in his first life, courtesy of his utter nonchalance regarding his own well-being and the simple fact that dying from food poisoning is extremely unlikely for someone without some kind of pre-existing condition.
The Grandmaster of Demonic Cultivation: Mo Dao Zu Shi fanfic By Any Other Name invokes this with post-resurrection Wei Wuxian, who as a result of a botched summoning ritual is physically frail and constantly in pain. It lasts until they do another ritual on the correct celestial alignment. Even then Wei Wuxian's health is fragile, risking overexerting himself at wrong moments.
In the first story, during her Break the Cutie sequence, she has been plagued with colds, depression, and various other ailments, even lack of sleep and appetite, requiring her to take so many medications to no avail. Ultimately, she was hospitalized and the doctors did all they really could. She becomes this once again post age-regression becomes this once again in Gensokyo 20XXV, after living in the Forbiddens for some time, and it was noted that she has been sick off and on.
In 20XXI, Eirin, due to stress, was starting to become this, developing some anemic condition as a result of stress, but she commits suicide (and succeeds) before it becomes an issue
In 20XXII, Keine is revealed to have died of an illness consisting of "pneumonia and lumps in her lungs", Aya dies of heart-attack caused by illness and stress, and Koishi started to become this.
In 20XXIII and later on 20XXIV, after the nukes are dropped, Sakuya dies of radiation sickness.
In 20XXIV, Reimu's mother, Reiko, dies of a long-term illness. Earlier, she was ill with radiation sickness before it dissipated.
From 20XXV, we have Yume Ni, in the aftermath of Chen's "The Reason You Suck" Speech. From what is said, she has a chronic illness and later on passed away. Later one, we have Ran, although, unlike the aforementioned Yume Ni, her illness isn't terminal, although it is hindering
In a side story, Reimu's Illness, because of her unstable (it is described as being up and down) health, Reimu falls ill with tuberculosis, human enterovirus 68, and viral meningitis and she is so sick that she's had to be transferred to a hospital in the city and be placed in something called an isolation box (which is akin to an incubator for premature infants).
Doing It Right This Time: As in the original timeline, Sakura got hurt during the battle against Sachiel. However her wounds were less severe and her arm was put in a cast (rather her legs) thanks to Touji trying to keep her out of danger.
In the story It's not the Raptor DNA, Elise's sister was this towards the end of her life. According to Dr. Gerry Harding, she was born with a weak skeleton, and couldn't have survived past a certain age. Elise's attempts to help her actually backfire and worsen things in the long run.
From Kill la Kill AU, we have Satsuki, as a result of stress taking a toll on her health. The first time is when she had appendicitis and the second time is when she has Tuberculosis. Later, in comic 41, we have this Ryuuko, after she makes a wish to be the sick one instead of Sats. This is also the subject of a more recent fanfiction where Ryuuko becomes ill and has to be hospitalized for tests. Although, as of chapter 9, it was revealed she had some unknown illness, which turned into septicemia, the which would have and probably could have killed her.
Ruby Rose in the RWBYHenceforward AU was stabbed straight through her spine with Blake Belladonna's Gambol Shroud. Though she narrowly avoids death and paralysis, this inhibits her physical capabilities to an extreme degree. Her reflexes are dulled, she can only lift half her body weight, and she must avoid any more physical trauma if for fear of the damage worsening. This keeps her from achieving her dream of becoming an official huntress and forces her to heavily redesign her weapon and fighting style to manage any sort of combat effectiveness. Funnily enough, though, is she probably worries about her condition the least and still makes a few reckless moves similar to those she would make in her youth (though she has attained a new sort of eerie grace in her fighting that she never had before). Her partner and possible lover, Weiss Schnee, is the one constantly keeping note of Ruby's condition and doing everything she can to make sure Ruby keeps from overdoing it.
StellaSqualo from the KHR fanfic Polaris is an example of this. She has some unspecified disease that breaks her body down gradually from birth, and she uses Mist Flame illusions for most of her life to keep her body from failing. When she does die, and the illusion breaks, her body is a rotting corpse with maggots in her flesh and exposed bone.
This is implied to be the case with Ryuuko, in the fic titled stop, come back, stay with me - (please) considering she was hospitalized for something and required a surgery.
From Maim de Maim, as we find out in one chapter, Ragyo's mother who suffered from routine staff infections that was mentioned to have physically aged her "worse than an average chain smoker". Said staph infections were due to her life fiber prosthesis not syncing with her body properly. Ragyo is still torn up about that.
In the fanfic Endless Numbered Days, there is Satsuki and her lung cancer. The cancer progresses to terminal.
Much of the drama in The Outside comes from Satsuki being this, as, much of the reason she's agoraphobic because of a triggered asthma attack when she was a 12, which almost killed her, on top her not being born healthy to start with, leading to her to confine Ryuuko indoors for about nine years (which in turn makes the latter more prone to illness). In the end, due to this, her life expectancy is shorter that Ryuuko's.
In crossover fanfiction The Institute Saga, Illyana Rasputin's need for medical treatment is why her brother Colossus works for Magneto. She gets cured by means of Kryptonian medical tech.
Tortoiseshell toms are often sickly. Emberpaw of Little Fires has always been more sickly than other cats and she almost died at two months old.
Stumble is a Harvest Moon oneshot where Elli is Secretly Dying of a genetic deteriorating disease. She became a nurse to cure her grandmother of the disease, but couldn't in time. Trent and Elli end up getting married and have a child, but Elli dies when their daughter is four months old.
Blind Courage: After being poisoned at age seventeen, Zelda still suffers from the side-effects years later. She frequently feels fatigue, coughs a lot, has a lot of headaches, and falls into sleeping spells. Her body also can't handle infections as well as she used to.
In Bitter Repercussions, Fireheart tells Sandstorm and Bluestar of a sickly older cat he knew as a kit. Lavender died of her illness after a few weeks of meeting him.
In Lonely Kagamin, Konata inherited her mother's fragile health, and becomes borderline-suicidal after hearing she'll die if she bears a child.
Us and Them: Aeris and Sephiroth's first daughter, Remielle, is born with a vestigial wing on her back. Since it's little more than a deformed limb that's literally skin and bone, it becomes prone to all sorts of infections, causing Remi to get sick on a fairly regular basis. This is one of the reasons she eventually has it removed when she's fifteen.
Sixes and Sevens: Dottie is this after the zero matter tries to take over her body, and Emily subsequently rallying Ivan's crew to try and get it out of her. Afterwards she needs Emily's help to stay steady, and Sousa compares her to a wet cat.
Mixed Examples
In Lucky For Hope, we have this with several characters: Mako, who is slowly dying from DTR (Deteriorating Tremor Rebounds); Gamagoori who has a form of leukemia (which doesn't have any obvious symptoms); Nonon, who, although not sick per se, is suffering the side effects of clinical trials; and Uzu, who seems to either be ill with something or the treatment thereof affected his breathing and eyes (his eyes are described as "smoky" and he has breathing tubes in his nose).
In Distance, we have Sakura, who is ill with leukemia, and Deidara, who is dying of a terminal illness.
The Phineas and Ferb collabfic Sick in Bed, in which Phineas and Isabella catch the H1N1 bird flu, and are bedridden together for the whole story.
In Dangerverse, Danger and Harry both have lupus as a result of having a version of the werewolf curse that does not result in actually turning into a werewolf. It' s not usually an issue, but there have been instances where they have gotten extremely ill.
In the first installment of Skyhold Academy Yearbook, it's eventually revealed that the real reason Hawke has taken a sabbatical is because Bethany has the potentially fatal North Llomerryn virus. Gender flipped with Cullen, who has a touchy heart condition from his time in the military. In both cases, the affected characters spend the rest of the series dealing with the permanent effects of their conditions.
Jill has a heart condition and wasn't expected to live to adulthood. Barely a season after becoming a farmer, she is hospitalized when her heart takes a turn for the worse. She is eventually given Jamie's heart and survives.
Jamie's mother died of a genetic illness that gave her an sudden heart attack.
Jamie dies of the same illness that killed his mother. It was brought on by stress from when Jamie thought he had caused Jill to have a fatal heart attack.
Zuko was "sickly" for most of his childhood, making him appear even weaker in comparison to Azula in his father's eyes.