(I don't yet have 50 rep points to comment on, and build upon, the accepted answer; otherwise I would, sorry!)
You can embed MongoDB in your OEM solution but there are two things to consider:
It is written in C++, so if you are coding in a different language you might need to write a wrapper that launchers the database process separately.
MongoDB is licensed under Gnu AGPL-3.0 which is a copy left server license. The accepted answer, and the Google group quote, both correctly state that this would normally force you to also be AGPL licensed. However, they MongoDb states that the intention of the license is to allow refinements to their code to be submitted back, and that your product will remain separate. This makes me think that the normal copy left rules don't apply.
The goal of the server license is to require that enhancements to MongoDB be released to the community. Traditional GPL often does not achieve this anymore as a huge amount of software runs in the cloud. For example, Google has no obligation to release their improvements to the MySQL kernel – if they do they are being nice.
To make the above practical, we promise that your client application which uses the database is a separate work. To facilitate this, the mongodb.org supported drivers (the part you link with your application) are released under Apache license, which is copyleft free. Note: if you would like a signed letter asserting the above promise please request via email.
Source: http://www.mongodb.org/display/DOCS/Licensing