I have vinyl tile floor in my basement workshop and 40% of tiles peeled up or
otherwise broke and I concern about tiles contained asbestos.

Relax, even if the tile does contain asbestos its not in the same state
as the type that causes health problems, which would be the stuff used
for insulation of steam pipes and boilers, or that stuff seen floating
around the air at Johns-Manville for fifty years.

Also, if the tile were really a health issue you would not be allowed
to dispose of it in your local landfill, which you are in NJ. I'm in
the Morris area and I've left more than a few bags at the curb, not a
problem.

HD sells floor scrapers that can be used to break and lift up the tile.
Then you shovel it into a garbage can containing a thick mil garbage
bag. Before the bag gets too heavy tie it off and haul it away.
Repeat.

For extra precaution set up good cross ventilation in the area you are
working in. Turn off any forced air units and put plastic over the
entryways. Wear a filtered breathing mask. Wet the area you are
working in thoroughly with a spray bottle. I doubt handling this job
will result in a slow and agonizing death.