If I play Erika's Invitation and put a Basic Pokemon onto my opponent's Bench would they put two damage counters on that Basic if I have Gapejaw Bog in play?
Yes, since the Pokemon is coming from their hand onto the Bench Gapejaw Bog places 2 damage counters onto it.
Source: TPCi Rules Team (2023-11-09)

Relevant cards

Note: All applicable cards may not be listed below.
SV35_160
SV35_160
SWSH10_142
SWSH10_142