Is there a law that says motels must take vehicle information from guests checking in?

This is not a law.  Instead, it is typically a policy that
is observed by motels in order to safeguard
themselves.

Motels typically need to worry about people
stealing from them, damaging rooms, or committing crimes while on their property.  In
order to try to protect themselves, motels often ask for the license plate numbers of
guests' cars.  This serves a number of
purposes:

• It can prevent people from using motel
  parking lots as a place to park for free.


• It can help the
  motel track down a guest who damages a room or steals from
  it.


• By asking for license plate numbers, motels can deter
  criminals from using the motels for criminal activity such as prostitution or drug
  sales.