IMO, that's a bad idea. Making them write the receive piece gives them lots of access to your database and they may not implement it in a secure way. Remember.. this will be sitting on the Internet so you need to make sure the security is there. Also, if they implement it, you may not get the code. If the next customer comes along and needs something slightly different, you're back to square one.

WebAPI is pretty straight forward, especially if you've done anything with MVC.