Am I allowed to use Gossifleur's "Call for Family" attack if my Bench is already full?
You can use the attack, but since it's public knowledge that you can't put any Pokemon onto your Bench the entire effect of the attack fails; you can't even search your deck.
Source: Supreme Victors FAQ (2009-08-27), PUI Rules Team (2009-08-27), PUI Rules Team (2020-05-07)

Relevant cards

Note: All applicable cards may not be listed below.
HGSS2_34
HGSS3_48
HSP_HGSS17
HSP_HGSS17