I live in Parkersburg West Virginia. I have no lease because my grandmother owned the property, which she recently sold to my cousin. I gave 30 days notice and moved to another location. My cousin now says he will sue me in court if I do not steam clean the carpets. Can he do this?

If the carpets were left dirty, he can withhold from your deposit the cost for cleaning the apartment.
You can defend it by showing it was normal wear and tear or not dirty.

Can he sue you, of course, can he win, that will be up to the judge

If there was a security deposit, the cost could be deducted from that. If there wasn't one, then he can sue you for the cost. I would do it. Its not that expensive.