Music licensing is bonkers but AFAIR (at least in the UK) I think you're allowed to do covers without explicit permission[1] - you'll have to give the original writers/composers the appropriate share of any money you make.
[1] Which is why you (used to?) get, e.g., supermarkets playing covers of songs rather than the originals because it's cheaper.
[1] Things like "was it on the radio or a TV show or a live performance or a recording? who was the composer? which licensing region was it in?" etc.