There are many sources of vibration that you will need to systematically go through to reduce your problem. You can’t eliminate vibration entirely, but finding the source can significantly reduce it. Possible sources are lack of proper preload, bad bearings, worn pulley/shaft fit, motor unbalance, and bad belts. There are probably others, but that is off the top of my head having a similar lathe a number of years ago.
First step is to look for anything loose. If any bolts have come loose, that could cause a vibration problem.
Since it runs fine at other speeds, it is likely not in the motor, but can’t hurt to check the vibration. Check the motor by removing the belts and see if you have the same vibration.
If that is ok, check preload on your spindle and adjust as necessary. If that is ok, see if there a problem with your belt. Belts have a natural frequency and if they are going bad, or too loose, it could change the speed at which they vibrate. If the belt looks questionable, replace it.
See if the pulleys are loose on the shafts, or pulleys with bearings/bushings are loose. They should not have any play.
Check the bearings as explained above, if they are wearing out, you can experience vibration at certain speeds you didn’t see before.
There is a lever which engages the drive system, if the spring is broken or weak, it could cause a vibration problem. Check this by engaging the drive with the machine powered down, and see if the belt feels loose. If you can easily move the belt with vey little force, the spring is likely bad. There should be some resistance to deflecting it.
The last thing is you need more mass for your stand to reduce vibration. Everything has a natural frequency at which it vibrates and by adding more mass, or stiffness, you change the natural frequency. If everything above is ok, you might just need to make your stand heavier or more rigid.