1999 Jeep Cherokee steering wheel Vibration

Assuming by vibration you mean shimmy. First thought is you have a tire out of round or tread creep. Or, it may be a front drive axle joint trying to bind up.

Question off the top of my head. Did you have this wobble before you put the new tires on. I sold brand name tires out of my shop and on a "very rare" occasion I would have one go bad within a month or two.
ASSUMING, your Cherokee is 4whl drive, I'm still curious about front axle joints.

Check also the track bar, this runs from the front axle to the frame just under the radiator, and the ball end at the radiator is the weak point.

Stuart, you're hot on the trail! There are 2 joints on each shaft. The inboard joints rarely go bad 'cause they don't flex as much as the outboard. Outboard joints flex more due to steering.
To check: With the vehicle in park, lift one wheel at a time off the ground. Turn the wheel back and forth by hand while watching the drive axle shaft. If there is "any" delay or "flop" between the movement of the wheel and the axle shaft, the joint is bad. You do the same test to check the inboard joint but, like I said, the inboards rarely go bad unless the vehicle has seen a lot of heavy 4-wheeling. There are "rebuild" kits available but they can be difficult to install. Rebuilt shafts are the most popular because they come both new joints. Any good parts store will have a listing for them. You can replace the axles your self but it takes some special tools. Hope this helps.