I’m very new to speed running and would love to try The Legend Of Zelda. But my NES stopped working awhile ago. People have been giving me mixed messages, am I allowed to use emulators, and if I am can I use Retroarch or do I have to use something specific?

And if all of that is ok do I just normally submit my runs or do I have to do something special because I’m on an emulator

you can use emulators but there's a lot of them that aren't allowed. If you intend to use one, I recommend FCEUX, with Nestopia or Bizhawk being nice alternatives. I also recommend showing your inputs on stream, though that's optional. Otherwise have at it!

Mesen and puNES are currently regarded by multiple resources as the most accurate emulators.

https:/​/​emulation.​gametechwiki.​com/​index.​php/​Nintendo_Entertainment_System_emulators
http:/​/​tasvideos.​org/​EmulatorResources/​NESAccuracyTests.​html

FCEUX doesn't rank well, but it's still accurate enough to be accepted.

Nesticle is inaccurate.

Pick an accurate emulator and if the program has any input lag reduction or predictive computation (e.g. RunAhead) or VSync type features they should be disabled because they cause inaccurate, sped up emulation.