I'm kind of new to speedrunning, and I was wanting to start out with Super Mario Bros. 3. I'm just wondering if it's OK for me to use an Apk to play on android or not. It is the only way that I can play the game, and I've tested it and it's the exact same as the NES version.

You’ll need to ask the moderators of the game. I don’t see Android listed as a platform option unless it’s considered an NES emulator, but the mods will be able to confirm one way or the other for you.

While I'm not a member of the community, I'm going to guess this won't be allowed. I don't know of any mobile emulators which are accurate (exact) enough for speedrun purposes. There are strong emulators available on computers, however.

I would run it under an emulator instead. Most old school games have an emulator area for runs. But you would have to check if the game allows it or ask a moderator.