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Does anyone have a tire and wheel they want to get rid of cheap in the Houston area? I'm thinking about purchasing a skeet launcher and want to weld it to a wheel. I live in the Clear Lake area.

try your local body shops

Go to a junk yard...they may pay you to get it off their property.

I recall mounting a thrower to a tire. I used the spare from my car. The goal was I could shoot by myself. I thought surely it would hold. The force of the arm threw the full sized tire/rim around. I had to drive rods through the lug holes into the ground to hold it in place anyway. Kinda defeated the purpose of the "portable' mode.
But-- YMMV--

Yes--junkyard for tire/rim.

A very long time ago a friend's father rigged something like that up so that he could shoot clay pigeons in his back yard (it was a rural area) and decided that he would fill the tire with water so that when he was done shooting he could empty the tire and save weight during storage.

You should have seen it when he pulled the valve stem to empty the tire the first time. This was in the days before tubeless tires were the norm, and his well had a head pressure of something on the order of 120 to 150psi. Household pressure should max around 80 with 40 to 60 being closer to normal, and even fire departments don't push it that high (120) for manned hose lines. He cobbled together a hose fitting to air fitting device and filled the tire with a hose directly off the well.

The stem and stem removing tool were never seen again, they made it the 200 feet to the railroad tracks and beyond.