In general the WebAPI controller suppose to be session less, the cart in Litium is stored in session. This is handled in the accelerators Global.asax file and turning on session state behavior for all request starting with /api/. Make a breakpoint in that method to see that the request if turning on the session.
Ah thats why, i have switched from /api/ to /json/ so i just need to add /json?
Edit: Worked like a charm, but if _requestModelAccessor.RequestModel.Cart.Currency will write to the session, should we use something else to get the Currency?