Bouncy Cart Tires
 

I have a Streetrod that has "TrailerKing" tires on it. They are quite a bit out
of round and cause the cart to jiggle up and down a lot...is this a usual
problem with these tires and what is a good replacement brand & place to purchase them!

This may sound really stupid. Try taking the tires off then putting they back on. When they install the tires, the don't tighten the lugs properly. I have had this isue.

Do these tires and wheels require balancing? Could possibly cause a symptom like this.

I know this will sound strange, as it did to me when told to try it. I have my tires at 28 lbs air due to hopping at certain speeds. I was told to try it, I did and it worked.

Air Pressure
 

Another possibility...If you drive the car very infrequently, tires can get a "flat spot" from sitting on the same area all of the time. Then when you first drive it, it is bumpy for a while until the tires adjust.

I was advised to keep 50 pounds of air in the tires while my "summer" car is in storage and then adjust the pressure when I intend to use it regularly. It does not harm the tires and eliminates the "flat spotting" that can occur.