I have a 1991 Jeep Cherokee that has a vibration at 55mph, goes away at 60mph and returns at 65mph. The front end does not shake but it is very annoying. All other driving speeds are fine.

Inspect both tires and wheels making sure they are not bent, a good sign will be a bad scuff marks on the outer rims lip. sometime balance weight can fall off causing it to be out of balance, have it re-balance. check for defect and tires wears it will give you an idea if its out alignments. check tie rod bushing making sure is not worn and wheel bearing will usually make some noise but check it anyway.

Sounds like the tyres are deformed with the sun ,or you just need a balance hope its the later its cheaper

Check your tire wear,have you changed tires on the vehicle if this started after a tire rotation it could be a pulled steel-belt/tread separation or from side to side rotation of tires, but normally that would create a pull to one side, if not check your CV joints if they havn't been replaced have them checked they can cause a vibe, it still sounds like a tire problem,chk tires for tread separation,steel belt stress,etc.have a tire co do a "run-out" gauge on the tires to chk for the above condidions