there's a few things this could be
One other thing to check for which isn't covered by a balance check is out of round or bent wheels, if you use a dial gauge on the axle to measure the run out on the wheel rim this will ensure thatthe wheels are round and true
Worn suspension joints - jack the car up so the wheel is off the ground (you’ll need to do this for all 4 corners) and shake the tyre from top to bottom and from side to side (with someone holding the steering wheel on the front) there should be no play in either direction. Any play in 1 direction will usually require a joint to be replaced, if there’s play in both directions then the wheel bearing may be at fault. If theres play only up and down its the strut top mount - to find this one you have to shake the suspension strut it self up and down with the wheel off the ground - so you may need to remove the wheel to check this one.
Now with the car jacked up on that side with the wheel about 2 inches off the ground use a steel bar under the wheel and lever the wheel up and down slightly, you should be able to look at the end of the axle and see if the suspension arm is moving up and down separately to the axle.If so replace the bush
To feel if there's a worn strut top bearing then lightly hold the spring by hand and have your assistant move the steering from left to right with the wheel on the ground - a worn bearing will feel 'notchy' through the spring. The wheel needs to be on the ground for the test to load up the bearing
Worn drive shaft joint - typically these won't produce any vibration unless badly damaged but they can be noisy or 'clonk' for a long time beforehand, turn the steering to full lock and look at the drive-shaft rubber boot there should be no rips or tears in it and the suspension should be clean and dry of any grease. If you reverse at a reasonable speed with the steering on full lock a worn joint will 'click'.
Worn mountings - check the tightness of the engine and gearbox mounts and their condition - this is best done by using a jack to take the weight of the engine off the mounts and seeing if any cracks or splits open up in the rubber.Also check the tightness of the mounts bolts
Propshaft : then I'd suspect that the propshaft either has a worn joint or the centre bearing is worn or its out of balance
you can easily check for for any slack in the propshaftjoints by twisting either half by hand and feeling for any play as there should be virtually none
the centre bearing is quite loosely mounted so this can be moved normally but if you can run the car in gear with the wheels off the ground and then feel the centre bearing housing for vibrationif there's any then consider replacing the unit
and then look along the propshaft for any signs of a missing balance weight as these take the form of little 3/4" squares that are welded onto the shaft - as they are only tacked on they can sometimes come loose and be lost resulting in the shaft being out of balance - if a joint is worn this can also shake them loose
so if a joint is replaced its good practice to have the shaft re-balanced afterwards