Dennis,
Travelling from Rome to the Cinque Terre for one night is not an ideal trip, as it will take 4-6 hours each way by train (depending on which train you use).
In summertime and especially in Vernazza, the crowds most likely WILL be brutal (I know that from first-hand experience). I've found that the "tourist crush" there seems to be more intense than the other towns. While it may be possible, I very much doubt that you'll be able to "show up and find a place to spend the night without booking prior."
IMO, one night will not be enough as it will be late afternoon before you arrive, which will only allow one evening for a very brief glimpse of perhaps one or two of the towns. You'll have to be out of your lodgings by about 10:00 the following morning, so hardly worth 12 hours of train travel to spend about six hours looking around.