Thanks for the extra info :-) I haven't tested it out, but there's a strong liklihood you could follow the model used by non-web solutions for Raygun. For example, here's the setup for a console app:
So you could new up a RaygunClient, include the API key directly. The only limitation (and this is similar to Lambda, but I haven't researched it deeply) is that I'm not sure there is a global unhandled exception event to hook onto. Therefore you'll likely want to add a try/catch block at a wide scope, and call the Raygun send event with the exception object from the catch handler.
I'll still have the team take a deeper look and see if there's a better way though Jim :-)
I hope that helps,