For speedrunning old games that exist on free websites, how is this uniformly controlled? For example, it would be really difficult to capture the screen for Super Mario Land on the GBA due to it being such a small screen...can it be played on any of those "retro gaming" websites (e.g. https://arcadespot.com/game/super-mario-land/) and captured on the PC for submission?

Sorry if this is a dumb question, I'm a little new to this :)

You need to ask the mods of the game, but those sites are definitely running emulation. However if the emulator isn’t known, then you may find it’s not allowed. But that’s for the mods to answer that definitively.