Wouldn't it curl up and sift from walking or moving furniture around if it's not secured? And what would prevent the tenants from not rolling up themselves to get to nicer floor underneath.
I would also imagine an apartment with all vinyl flooring wouldn't be as nice as one with carpeting. It would bring down the visual appeal and may affect the quality of tenants it will attract especially if you are not adjusting your rental price.
Have you looked into commercial carpeting? It may be a bit more expensive but it is a lot more durable than normal carpets. It's meant to take a licking and have high traffic.