The All American Pups series: Fritz, who freaks out at everything from the local cats to costumed humans. He's friendly and the other puppies like him (aside from some exasperation), but he tends to be thought of as the baby of the group.
Lemony Snicket himself in A Series of Unfortunate Events. In nearly every book, while narrating some terrifying situation, he comments that, had he been in the Beaudelaire's place, he would have been unable to go on and would have instead run away in terror, dissolved into helpless tears, etc.
'Besides...where Rincewind went’ – he lowered his voice – ‘trouble followed behind.’ Ridcully noticed that the wizards drew a little closer together. ‘Sounds all right to me,’ he said. ‘Best place for trouble, behind. You certainly don’t want it in front.'
To some extent, Fred Colon and Nobby Nobbs, though they've seen trouble before and usually square up to it if there's no other option, or Vimes needs them to do so. But mostly, "When the time came, he [Nobby] would not be found wanting. He would not be found at all."
"He had always thought heroes did heroic things for God and Country and mother's apple pie. He never thought they might did them because they might be yelled at if they didn't." (Actually, Colon isn't so concerned with being yelled at in that scene as with having Vetinari raise his eyebrow at him.)
In Castle in the Air, the sequel to Howl's Moving Castle, Sophie describes Howl as being cowardly (as well as sly and selfish and vain as a peacock). When Abdullah comments that she seems strangely proud of Howl's vices, she states that she's just describing him. (She does really love him, including his bad traits.)
Horace Slughorn of Harry Potter prefers to take the easy option over a direct confrontation with the enemy, but proves his inner bravery when he rallies reinforcements in Hogsmeade during the Battle of Hogwarts and leads the Slytherins into battle after previously evacuating them, not to mention going toe to toe with LORD FREAKING VOLDEMORT (albeit with the help of two other powerful wizards). And even before that, he's still a likeable guy.
Hagrid's gigantic boarhound Fang proves in Harry Potter and the Philosopher's Stone that though he's a sweetheart of a dog even when he's with people he likes he will flee when faced with something that scares him, like a cloaked figure drinking from a unicorn's corpse.
Mundungus Fletcher's cowardice and dishonesty are generally played for laughs and he remains a member of the Order of the Phoenix in spite of them. There's never any implication that he would actively betray the good guys to Voldemort, it's just that he also can't be relied upon not to take off when danger looms.
Pierre Gringoire from The Hunchback of Notre Dame. He tries to do his best to help save Esmeralda, but when his own neck is at stake he decides he'd rather not.
Piglet of Winnie-the-Pooh is almost certainly this, a timid stammering Neat Freak who fears every superstition and mythical animal his friends make up. The Disney adapted version dials this Up to Eleven (along with adding Lumpy to the list as well).
The title character of the story "Philbert the Fearful" in The Practical Princess and Other Liberating Fairy Tales, whose Catch-Phrase is "I'm the only one of me I have", and who is much more sympathetic than the other knights, all portrayed as Fearless Fools.
The space opera BALADA: A Symphony of Eternity series has Metternich per Pelasgiamus, he's a coward more in a sense that he's a civilian who was drafted into the fleet against his will and is just trying to survive; the only problem is that when he survives a battle, he looks like a great hero and so is forced into even nastier scraps.
Edgedancer (a novella of The Stormlight Archive): Wyndle is always on the verge of nervous breakdwon and often argues that he and Lift should really just leave town and the Big Bad behind so that they won't have a ton of nasty stuff happen to them. That being said, his pestering is oddly adorable, and he acts as Lift's Morality Pet once he realizes they're not leaving.
The Unknown Soldier has Pvt. Riitaoja. Riitaoja is panicky and wails like a child whenever anything dangerous happens. However, the rest of his platoon find his childlike nature endearing, and don't mock him because they all know that everyone is just as scared, and the only difference between them and Riitaoja is that they have learned to grit their teeth and bear it.