The Vampire Incident is an event loosely described in the book Perfect Memento in Strict Sense, and it took place after Mystic Square and before the Embodiment of Scarlet Devil. In it, Remilia gathered followers and rampaged across Gensokyo, before being stopped "through the efforts of the most powerful youkai". In the aftermath, the Spell Card rules, which among other things guarantee Non-Lethal Warfare, were created, and the first incident to see them in use was the Scarlet Mist Incident from EoSD.
Here's the thing. The creation of said rules seems to imply the severity of the incident, and by extension that lives were lost. It's also implied that Reimu had no involvement in solving the incident (only in settling the aftermath, as she was involved in implementing the Spell Card rules), since it specifically said that Remilia was defeated by youkai, no mention of any humans, let alone the Hakurei Shrine Maiden.
Mizuchi might be the vengeful spirit of a human who was killed in the Vampire Incident, and blames her death on Reimu for not having done her job at that time, hence her grudge.
It's also why her attacks have targeted people across several Windows games in chronological order. She's going through the incidents that took place after her death, and is probably pissed that Reimu has gone back to being consistent at bothering to do her job "too little too late". Obviously the out-of-universe reason she didn't start her attacks with characters from the PC-98 games is the Soft Reboot of the series after the leap to Windows, but the aforementioned point would be the In-Universe explanation.
One possible answer to her grudge is simply that the creation of the Great Hakurei Barrier was not as clean or family friendly as we have been lead to believe. Perhaps the whole Miyadeguchi family was sacrificed or even betrayed (or at least betrayed from Mizuchi's point of view) which served as the source of her hatred.
