There are a few things you should know about releasing a cover.
Good news, you don't need permission to release a cover as long as it has already been released by the original copyright owner.
Releasing a cover either on a physical product, like a CD or vinyl, or online, requires payment of royalties. If you're represented by a label, they will likely already be covered by a licensing agreement. Check with them to see if you're covered.
Licensing for physical products, like CD or vinyl
For physical releases you need something called an audio manufacture licence, which is available through APRA AMCOS.
Licensing for online release
For online platforms that are already covered by a licensing agreement, like Spotify, you don't need a licence. However, if you plan to release it through an unlicensed platform like your own website, you'll need a licence.