Will removing supporting cabinets cause table top to fall?
No way to tell if the top would fall without knowing how it was fastened to the wall. You're right that if you add the vinyl and then try to slide the cabinets back in, assuming the top didn't fall, they won't fit. If you tried to force them in, you'd damage the vinyl. You could get a belt sander and remove some material from the bottom of the cabinet. 
Personally, I think your best bet would be to remove the tops and take your time and do the job. Reinstall the cabinets on the vinyl and then install the top, secure and caulk the sides. good luck and stay safe out there.