If I have 4 Togekiss with "Wonder Kiss" Ability in play when my opponent's Active Pokemon is Knocked Out, do I get to flip 4 times even though I can only take one extra prize, assuming one of the flips is heads?

No, you only get to flip once. "The effect of Wonder Kiss does not stack" means that you can only use the Ability once regardless of how many Togekiss are in play. The coin flip is part of using the Ability, so you can't "stack" the flips.

Relevant cards

SV8_72