Problem with Process Builder Invocable Method and Webservice Call out

Apex can be annoying like that: limitations in termos of callouts and DML. What we've done to work around that is to make callout in the future method. Your invocable would call the future and pass the map. The future would construct the request and send it out. Your future should be annotated with @future(callout=true) to be able to make callouts.