In case anyone else scrolls down to here looking for an answer. As of today there are two options for Cosmos DB with the MongoDB API. These are "Azure Cosmos DB for MongoDB (vCore)" and "Azure Cosmos DB for MongoDB (RU)"
The "RU" instance has a data explorer within the browser.
The "vCore" does not.
You can still use GUI like Studio 3T to login, view documents, collections and write queries