If a Pokemon normally has no Weakness, does Vileplume's "Allergy Panic" Ability give it a x4 Weakness instead?
No, if a Pokemon doesn't have Weakness, you cannot *apply* any Weakness multiplier to it at all.
Source: Boundaries Crossed FAQ (2012-11-08), TPCi Rules Team (2012-11-08)

Relevant cards

Note: All applicable cards may not be listed below.
BW7_3