Welcome to the forum!
As part of the troubleshooting procedure, try disengaging the drive to the lead screw. That will check out the gear train between the spindle and lead screw. If the vibration doesn't go away when the drive is disengaged the problem could very well be the spindle bearings, although the drag on the carriage (when driven) will load up the entire drive chain from the motor on. But you gotta start somewhere, right?
Based on its light use, it doesn't sound like the spindle bearings are worn (though one could just be plain defective). If the problem is a loose spindle, the spindle bearings on these need a pre-load, achieved by properly tightening the spindle nut on the change-gear side of the lathe. If you don't already have one you may need to make or buy a spanner wrench for that. The ad-hoc way to adjust it is to tighten down the nuts and run the motor awhile, then check to see how warm the headstock gets. You want that "Goldilocks" pre-load where the temperature rise isn't too much or too little.
I wish I could tell you more but even 10+ years into owning one I haven't needed to adjust the pre-load on my mini lathe, and no vibration unless I'm turning an unbalanced piece. FYI, Arceurotrade has a PDF on replacing spindle bearings you can consult for guidance.
Good Luck! And let us know what you find. I for one am curious.