For the merry-go-round problem, do the magnitudes of the position, velocity, and acceleration vectors change with time?

No

Explanation:

Although the direction of position, velocity or acceleration of an object in marry-go-round problem changes continuously, however the magnitude of the position, velocity and acceleration do remains the same. Marry-go-round is nothing but a machine found in fairs that turn round in circular motion. So, the laws of circular motion are applicable in it.