Trailer shudder

[Johnboy]

If you have a spare tire that you know is a good one. Put it on and take the trailer for a test drive. If that doesn't eliminate the problem swap the spare to the other side.
Unless you have a dual axle, then it will be a little more tedious to determine if it is a tire problem.
Doing that will give you the opportunity to check the wheel bearings for tightness and smooth rotation.

[DW]

Consider the possibility that the problem is in the tow vehicle. When loaded with a trailer something like a drive shaft failure could surface. Can you do a test tow with a borrowed trailer?

[ffishman]

Most likely a tire going bad, but could also be a bent axle

[REW]

Put on a new set of tires and see if the problem remains.