Asbestos is usually found in vinyl tile laid before the late 70s. I believe its use was banned in the US in 1978, so if you know your floor is newer than that you should be good.
If you want to be 100% sure you can get a test kit (I got one from Ace hardware for $20 or so). The kit comes with prepaid envelope, you mail in a sample of the flooring and they email you a report on its contents.
If you google asbestos flooring you'll find there's a ton of information on the subject.