Eaton Output bearing

It is more likely that the nut holding the yoke on is loose. The u-joint will need to be removed from the yoke to access the nut. Use a new nut and preferably torque the nut instead of just hammering it on with an impact wrench. Seal should also be replaced and check the yoke to see if it has a groove where it rides against the seal. If it is, then there are seals with a ring that goes over the yoke to provide a new surface.

 

I would start with torquing the existing nut. If the play in the yoke tightens up then pull the yoke and inspect it. Can't remember if you can pop the rearmost bearing out. If you can, give it a look over. If all good put the bearing back in, install a new seal and torque the yoke up with a new nut. Might save you from a lot of work

If torquing up the old nut doesn't take the play out, its time take it apart to see why.

 

Agree with both above. You can tighten it with a chisel without dropping the driveshaft

See if that eliminates your slack. If so, yes, new nut and seal will fix you up.

Personally, I bet the vibration is coming from further back

 

The rear bearing is a tight fit on the OP Shaft, but if the nuts been loose, the bearing is often easy to pull with the seal housing removed.

 

Thanks all of you guys and you are on the same page. I agree with tighten it with a chisel. If that took the slack out that is great but it will never stay. Pull it down, do the new seal ( wear ring ) nut (They are considered a one time use item) and go again.
You never said what transmission you have so I am assuming that is a twin counter shaft transmission. Those have two bearings facing each other like a wheel bearing in a car but have a spacer between them. If they have been ran loose long enough the races begin to spin in the housing. The housing is worn. No bearings races or nuts will fix that.