If the floor is just vinyl (like a 10 year old floor that's just worn out or ugly) then you don't need to worry about asbestos (unless there's linoleum underneath it.)
I've torn up floors before that were vinyl with a felt backing - the top layer comes up and the felt stays on the floor in patches. For that type of floor, try various chemical strippers to soften the adhesive. Go to a hardware store and read the back of an adhesive container. See what they recommend for removing it after it's dried.
Real linoleum doesn't have asbestos in it either (although the glue might) - Real linoleum is made of completely natural materials - hasn't been around for ages (early 50s?) - so it's doubtful that that's what you've got. (Although it's making a comeback.)
Is it one big sheet of vinyl flooring or individual, brittle tiles? (THOSE probably contain asbestos.)