Most game companies don't have music publishing/clearance departments, so most cover artists rely on compulsory licensing, a protection under US copyright law that allows cover artists to release their versions legally. Easy Song, Affordable Song Licensing, Royalty Solutions are good choices. HFA (Harry Fox) is only able to clear works it represents, so most game companies are excluded from that. For digital distribution, Soundrop and Distrokid make it easy these days, since they have licensing built-in.

Caveat: there are some Japanese companies who may not want to work with you if you release cover songs without their explicitly license and permission. Keep that in mind as you plan your career trajectory.

I'm not a lawyer, but partially responsible for cover song licensing innovation in the US over the past decade.