Best HTTP Method for Get or Create

I would use GET for this. Repeated calls to this end point will return the same resource, so it's still Idempotent.

A GET request expresses the user's intent to not have any side effects. Naturally, there will always be side effects on the server like log entries for example, but the important distinction here is whether the user had asked for a side effect or not.

Another reason to stay away from GET surfaces if you respond with the recommended 201 Created response for a request where the resource is being created on the server. The next request would result in a different response with status 200 OK and thus it cannot be cached as is usually the case with GET requests.

Instead, I would suggest to use PUT, which is defined as

The PUT method requests that the enclosed entity be stored under the supplied Request-URI. If the Request-URI refers to an already existing resource, the enclosed entity SHOULD be considered as a modified version of the one residing on the origin server. If the Request-URI does not point to an existing resource, and that URI is capable of being defined as a new resource by the requesting user agent, the origin server can create the resource with that URI.

If a new resource is created, the origin server MUST inform the user agent via the 201 (Created) response. If an existing resource is modified, either the 200 (OK) or 204 (No Content) response codes SHOULD be sent to indicate successful completion of the request. If the resource could not be created or modified with the Request-URI, an appropriate error response SHOULD be given that reflects the nature of the problem.

In the above form, it should be considered a "create or update" action.

To implement pure "get or create" you could respond with 409 Conflict in case an update would result in a different state.

However, especially if you are looking for idempotence, you might find that "create or update" semantics could actually be a better fit than "get or create". This depends heavily on the use case though.