If I draw Snorlax Doll when drawing cards for my opponent's Mulligan, can I play it to the Active position?
No, by that time, you have already placed an Active Pokemon. But you can still play it onto your Bench if there is room.
Source: TPCi Rules Team (2023-11-09)

Relevant cards

Note: All applicable cards may not be listed below.
SV4_175
SV4_175