Is it possible, and if so how, to use an Azure MongoDB as the backend for my Meteor Application. I have added the connection string from my database into the MONGO_URL variable
In your example, you're actually using DocumentDB with MongoDB compatibility. You're not using native MongoDB (nor is this native MongoDB as-a-service).
DocumentDB (even with MongoDB compat) does not provide an oplog. And since Meteor has a dependency on reading the oplog, you wouldn't be able to point Meteor at DocumentDB.
In your case, you'd need to either run native MongoDB on your own (e.g. in VMs) or take advantage of a 3rd-party MongoDB hosting solution which provides MongoDB support within the same region as your app. (ok, yes, you can run your app in a different region, but you'd see latency along with data egress charges).