I found a game I want to run that is for the PS1, PS2, and PS3. I did a little digging (not that much) and I couldn't find anything. Does anybody know any way to emulate a playstation game?

yes you can, but it depends on the game mate

This game https://www.speedrun.com/soko-ban_basic

only if a game has a native PlayStation release you can run it

I'd recommend checking with the community whether and which emulators are allowed for a given category. For some categories there are technical reasons to require a specific emulator, or even ban emulators under certain circumstances.

That being said, for PS1 in particular mednafen is a pretty accurate emulator and thus usually allowed (https://mednafen.github.io). IIRC the Bizhawk emulator (https://tasvideos.org/Bizhawk) uses mednafen under the hood for PS1 as well.

Another popular and fairly accurate emulator for PS1 is Duckstation, so worth looking into as well (https://www.duckstation.org).

For the PS2 I'm less certain but I believe PCSX2 remains the most popular and accurate emulator (https://pcsx2.net). Not sure if it's as ubiquitously accepted as with PS1 though.

I frankly have no idea if there are accurate enough emulators for the PS3 to be allowed for competitive speedrunning. But you can always ask the community for the game when in doubt (this also applies any kind of inquiries in general).

EDIT: Also bear in mind that for any given emulator a specific configuration might be required, and proof that you are using the correct configuration provided for any submitted runs. Again, the best option is to check with the game's community.

thanks guys

ps2 and ps3 speedruns generally do not accept emulator runs due to their inaccuracy. ps1 games limit to usually mednafen and duckstation and even then you need specific settings

Fun fact you can speedrun on an "illegal" emulator without submitting to the leaderboard if you just want to run the game.