Can I submit my game as a ROM file or does it have to be in .exe (downloaded with the emulator)

You can submit any format. It's recommended that you make it playable on the web for more people to play it, but it is not a requirement. A ROM would be more trustworthy than an exe.

Previously I've used a javascript emulator so I can submit a web version aswell as the rom. 

I used this emulator: https://github.com/chrismaltby/gb-studio/tree/develop/appData/js-emulator (seems like fork of https://github.com/taisel/GameBoy-Online )

Example: https://shadowsora211.itch.io/impossible-gameboy