This would have been a Carl Miler, Ben Schiff or Molina & Sons built kiddie coaster. All there manufacturers were nearly identical in design and difficult to tell apart without close-up pictures of the trains.
Many rides at Asbury Park were indoor at "Palace Casino and Amusements". The Ferris wheel was boarded from inside this facility for example. Asbury Park also had many typical boardwalk concessions and appears to be the case for this small roller coaster. It existed outdoors near the entrance to the "U-Pedal Boats".