If you’re an artist who would like to cover a track, we require that you hold or otherwise control all the rights of the music you upload. So you'll need to have a performance license (to cover streaming playback) and a mechanical license (to cover downloads) for any covers you upload to Bandcamp, which you can often obtain with the help of some third-party licensing organizations. Once you have those then you're set - but keep that paperwork on hand in case we need it for anything.
Check out this page for more information: https://bandcamp.com/terms_of_use#upload.