Raven's Store method is analogous to INSERT in SQL Server. There is no UPDATE, you just load the object from the database (which puts it under change tracking), change it, and then call SaveChanges.
It wouldn't be a good idea to tell Raven "Here's this object I got over HTTP, please update this document id with these contents" because then you would be giving up Raven's E-Tags that give you control over concurrency.
Depending on your situation, it may or may not be the best idea to send your Raven models directly to the client via WebAPI and then get them directly back. Whether or not you use the exact same classes or separate DTO type classes to communicate to the client, you could use AutoMapper to map the properties from the object coming in over WebAPI to the RavenDB model you load from the database.