I notice you’re using a sandbox domain. MG doesn’t allow delivery to anywhere except the addresses registered with it if you’re using the sandbox. If you’ve got a mismatch between the addresses MG thinks are OK for the sandbox and the address you’re trying to sign up, that’s a problem.
You probably want to switch from the sandbox to a real domain, asap.
For anyone else going down this road, one thing to add is that I needed to create a Mailgun API “sending key,” which is different from a regular ol’ API key. You can find it in the Mailgun sidebar in Send → Sending → Domain Settings → Sending Keys