Pokemaster is probably better fit to answer this, but in short, yeah possibly. He’d have measures in place to deal with activity spikes though — you’d need them for a franchise like this where new games and unexpected new reveals will jolt your usage suddenly.
However, people do nasty things such as what is called a ‘denial-of-service attack’ (DoS) where they indeed crash a server by sending it too many requests at once. It’s not something you can do without the technical know-how though, and any good site makes it hard for you to do it anyway.
I know Smogon and especially Showdown used to get these a lot some time ago, but I can’t imagine any motive to crash pokemondb.