People do it all the time (rarely for free) but the official answer is no.
Not what you wanted to hear, we know, but that is the honest answer.
We are not licensed to broker the shipment of household goods and, likewise, no car carrier that operates in the USA is licensed to transport them from state to state either. Despite what you might be being told by other car shipping companies you may speak with.
Remember, at the end of the day we're all salespeople, and the true answer to this question is not a great selling point.
You will hear a lot of companies tell you that you can put up to 100 pounds of items in the trunk, but that is not entirely true. That fact is that items of that amount are fairly common and the department of transportation is probably not going to split hairs and fine the trucker over items of that amount, provided they are not over their weight limit. They could fine them, however, if they see a vehicle stuffed full of personal items so the car carrier will most likely try to negotiate something with you to cover themselves against any costs they could incur. It's not something we can build into your contract though.
We have a short and helpful video on this topic in our user videos.