I guess I'm really not this extension's target audience, because somehow Mozilla managed to make me uninstall it in just a few minutes. First, they have forced telemetry. Okay, it's an early release, it's very basic information, they want to understand how they're doing - I don't like it, but I can understand it. Sets a wrong vibe, though - I had to check if it was from Microsoft and not Mozilla. ;-) Then, I figured…
Based on this I would assume they are using GCP Vertex AI as that's going to be WAY cheaper than rolling it all themselves and hosting the model on a GCP server instance. I would also assume they'd be using the gcloud SDK for Vertex AI/Model Garden, which I believe means they can't just provide for a different endpoint and payload shape if you had a service elsewhere.
Eitherway, at the (presumed) scale they'll probably also be using GCPs API management service, so I would expect further abstraction between what the extension is sending and what the model/Vertex AI expects as a payload. This means providing that kind of "bring your own endpoint" experience would require more bespoke build-out time.
BUT who knows? Maybe this is just straight up hitting the out-of-the-box GCP Vertex AI REST API directly from the extension like some hobby project.