You can obtain mechanical licenses through an agency:
Harry Fox Agency
Easy Song Licensing
Loudr was another place to do that. They were acquired by Spotify some time ago. There may be a link on the Spotify site.
But if this is simply for YouTube, you may not even need the license. If you see other covers of the song you're going to post, chances are the artist is fine with covers being done and they may simply claim the royalties rather than having YouTube take the song down.