I can't completely visualize how it works with a VPN in the middle, or whether the VPN provider needs to (not) do anything to allow it to work. i just need to try it out.
Discussion
The VPN just relays all your traffic including keet, keet is designed so you don't need to do anything for it to work, if you did it would be too difficult for people to adopt
Keet has a process of hole punching to make connections which is the key to its success and why they named the company holepunch
This is a great video on the tech and hole punching https://www.youtube.com/watch?v=QWWHKLyCFTU
There is connections that are more compatible with keet and they show as green in the keet profile page, there is yellow connections which require other peers to help but obviously still work