Yes, the wipers have to be operational in Texas. And the vehicle is required to have wipers. Some of the older vehicles only had one wiper and since that was standard equipment that is all that would be required.
But yours would have had two wipers so they must be operational.
Registering yours as an antique may limit your driving abilities though. May limit you to driving it for special events such as to and from parades, car shows and such.
I'm not really up to date on this though since mine is registered for every day use.
