You'll still need permission from the original copyright holders of a song before you can post your cover on SoundCloud, even if you provided the vocals and played all the instruments. After all, the song itself is still someone else's.
This is obviously not specific to SoundCloud. But then how on earth should a musician contact a band to ask for permission to do a YouTube bedroom cover?! CDBaby states that a sync license is needed but doesn't explain any typical way to approach the music publisher.
- If multiple music publishers are listed in the liner notes, does one administer all rights in a particular song, or does a reuser have to negotiate individually with each one?
- Do music publishers tend to require a recording artist to have the backing of an established label before granting a sync license?
(Based on a question by Mr. Boy on Music: Practice and Theory that was closed as off-topic and is too old to migrate. It was recommended in a comment to ask here on Law Stack Exchange instead.)