Like so many things mechanical, you've got to get in there and try stuff.
Can you remove the worm, just leaving the shaft with its gear in place, and then run it in its place to see whence the binding? (Seems to me that you could pull down on the shaft with your finger to partially simulate the string load.) Is the shaft properly constrained in the through hole on the tuner side and the blind hole on the other side, or are those holes worn into ellipses? What does the surface of the shaft look like at each bearing surface? Is it smooth and polished, or rough and galled up? Is the shaft bent? Is the gear square on the shaft?
I would have answers to all these questions before I start squirting lubricants around. Unless the shaft is properly constrained in two good round holes just a wee bit larger than the shaft ("close running fit" is what I would expect), all the bearing surfaces are smooth and polished, the shaft is straight and the gear square on the shaft (and there are probably some other considerations too) adding lubricant won't fix your issue and might make it worse. What is the material of the shaft, is it brass, wood, or something else?
[edit, I just took a look at my inexpensive individual-plate tuners of recent manufacture]
If you have ordinary individual-plate tuners like mine it looks like you could undo the gear from the shaft, pull the side plate (which holds the worm in place) then put the gear back on the shaft and see who it runs in its "bearings". Another thing that occurred to me is the total shaft length could be too long and its end bearing on the bottom of the blind hole on the far side of the pegbox. Is the stiffness of the tuner something that has recently developed, or has it always been like this?