It can't be done. "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.
As noted in the comments, copying the activated ability would work and let you keep more than one permanent tapped.