What emulators are allowed and for what console?

Ask the mods of the game.

Usually, anything old (N64 and earlier) will allow emulators because they tend to be accurate to real hardware. Just make sure you use an apporoved build/model - the game mods can provide you with approved models.

Anything more modern (say PS2 era and beyond) are usually banned or split out from a real console because those emulators tend to be very inaccurate.

Generally, specifications for emulators are usually in the rules for any given game.

Some are known to be inaccurate emulations, from speedrunslive.com:

SNES - ZSNES, snes9x 1.4x N64 - Project64 2.x GB/GBC - VBA, VBA-M GBA - VBA-M