It was a fake arm (or at least, it wasn't his arm).
I think the implication is that it was in the package Fester gave them. The only mention in the script:
In a final vengeful moment, Wednesday HACKS AT PUGSLEY'S
LEFT ARM, CUTTING IT OFF, SENDING A GUSHER OF BLOOD OUT
OF HIS STUMP. THE ARM DROPS TO THE STAGE AND BOUNCES OFF,
LANDING IN JUDGE WOMACK'S LAP.
You could interpret it as cutting his actual arm off, and then someone sewed it back on. The Addams films are full of implied maimings and things that would kill someone, but have no effect. Especially to Pugsly and Fester.
But I think the more obvious answer is they went beyond expectations in stage effects for a school play (and they already did with the choice of Hamlet).