Asbestos fibers can lead to deadly health problems. That is undeniable, however, the asbestos in vinyl floors and in its adhesives is categorized as "nonfriable" which means unless you grind/pulverize, etc, it's remaining within the matrix of it's containing material. It's not like the insulation wrapping a boiler.
There are certain buzz words for lawyers. Asbestos is one of them. Following safe removal practices, most places allow homeowners to remove vinyl floors themselves and some even consider it as regular solid waste.
The readers digest version is to keep the area where you are working wet with water and dawn dish soap using a garden sprayer, while wearing proper personal protection and keeping a positive air supply into the room and keeping it sealed off from other rooms. Double bag the trash and mark it as hazardous, asbestos. Use a shop vac with a hepa filter to clean up the slurry of water and adhesive you've scraped while keeping it wet.
You can read the complete instructions at The Resilient Floorcovering Institute of America Website at [www.rfci.com] under their technical information section. The pdf file is pretty large as it contains removal instructions for every possible situation, from tile over slab, to sheet over plywood, which includes removing the plywood with it or saving the plywood. Each situation is only about 8 pages of instructions written at about a 3rd grade reading level.
I suggest cleaning it real well, taking off any wax, etc., then to scratch it up
That advice WILL release fibers!
The tile on the floor now will act as a great crack suppressant membrane
WHAT? You are kidding, right? Seriously, you are kidding, right? Vinyl as an ANSI recognized crack suppression? When did you make the switch from duct tape?
Here in California you must be certified to remove Asbestos unless it is 100 square feet or less but still follow all regulations.
Hey Leasure, there in California, even fruit sold at a road side stand comes with warning labels

After 25 years of setting tile, stone and restoring old mosaics, my brother had to get out of there 3 years ago. He couldn't take it anymore.