Can Amber Prison prevent multiple permanents from untapping?
The only way it can be done is by copying the ability (such as through Rings of Brighthearth.  With just one copy of the ability, it works as follows:

"As long as" denotes the duration of the continuous effect. The effect ends when Amber Prison is untapped, and when it ends, there is nothing preventing the first creature from untapping.


  611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it
  (such as “until end of turn”). If no duration is stated, it lasts
  until the end of the game.


Even if Amber Prison is  untapped while its ability is on the stack and activated again, it wouldn't be possible, becase the effect from the first instance of the ability would not even start:


  611.2b Some continuous effects generated by the resolution of a spell or ability have durations worded “for as long as . . . .” If the “for
  as long as” duration never starts, or it ends before the moment the
  effect would first be applied, the effect does nothing. It doesn’t
  start and immediately stop again, and it doesn’t last forever.