First off, I too feel your pain. I had the unfortunate experience of developing in MagicFS/Focus a couple of years back, and we had the exact same request for relational query support. Why do people always want what they can't have?
Anyway, if the version of MUMPS you're using is anything like MagicFS/Focus and you have access to the file system which holds the "database" (flat files), then one possible avenue is:
Export the flat files to XML files. To do this, you'll have to manually emit the XML from the flat files using MUMPS or your language of choice. As painful as this sounds, MUMPS may be the way to go since you may not want to determine the current record manually.
Read in the XML using LINQ to XML
Run LINQ queries.
Granted, the first step is easier said than done, and may even be more difficult if you're trying to build the XML files on the fly. A variant to this would be to manage generation of the XML files like indexes, via a nightly server process or the like.
If you're only going to query in specific ways (i.e. I want to join Foo and Bar tables on ID, and that's all I want), I would consider instead pulling and caching that data into server-side C# collections, and skip querying altogether (or pull those across using WCF or like and then do your LINQ queries).