What are the private keys?. Sorry if that is a bone question. If it's the wallet address I sent it to then I have it. Annoyingly I can see on block chain the BTC are still there and not touched or claimed. That's crazy that this can happen.
The private key is what allows you to spend bitcoin. Your wallet software/service handles them for you and they are usually protected by your password. When you send bitcoin to someone else you can not reverse the transaction. Thats a fundamental bitcoin principle.
All you can do now is hope they are indeed legit and contact their support.