Most consumer internet service hosts only offer mainstream SQL databases like MySQL or PostgreSQL.
If you want to host your app online using MongoDB, you could look at
Heroku which provides
application hosting services aimed at developers. You can opt for
add-ons which include some MongoDB options, but you have to pay extra for these (the DB price will depend on how much data you want to host or transfer in/out).
Another option might be
MongoLab which appears to offer MongoDB hosting, including a free basic service.
Alternatively, you could look at Amazon's various services which may allow you to run a VM including MongoDB to support your application. The services and pricing structures here are pretty complicated, so take your time to figure out what you need and what it will cost. Amazon does offer a free tier of services for an initial 12 month period which might be enough to meet your needs.
You probably need to look carefully at the various options and figure out how much you want/need to pay to get the right level of service for your needs.