I know you can catch regular Pokémon by browsing here, but how do you get legendaries? I'd love a Lugia or Azelf.
It varies, there are some legendary eggs that are around the site. Those need you to make them your buddy for some time as you post, once they hit certain levels they then hatch! As for other legendaries, they come and go as they please, not being around year round. Lake Valor has its own Pokédex called the Valor dex which includes the usual roaming locations of the Pokémon in question. Its not the most up to date but you can find good information on a lot of Pokémon there. Here is the link to the page for the Valor dex: https://lakevalor.net/pages/pokeindex/
In case you want a certain mon but its location is not available in the Valor Dex, there's one subforum dedicated to update more info about which mon is in where. Here's the link: https://lakevalor.net/forums/valordex.278/