Though I'm aware of music bots for Discord, I simply post the youtube link for the ambient background music I picked and the players can play it however they like.
These are long, instrumental ambient tracks, usually, or, if they're short, they're practically seamlessly repeatable, and I don't vary them as often as I would in a live game. There's a "common" track that suits the setting's general mood, there's a "tension" track, and there's a "combat" track. Makes things simpler.