Can you stay in a hotel without a reservation?

Of course, unless a given hotel absolutely requires a pre-made reservation. If there is room in a hotel, it doesn't seem to make sense denying a 'walk-in' an opportunity to pay to stay in it. But traveling around without hotel reservations is very dangerous, and likely to cause a lot of trouble. If there is 'no room in the inn', not even a sweet baby from Bethlehem is likely to be accommodated. Make reservations.