If I'm not allowed to play a Supporter card for some reason, can I still use Mimikyu's "Impersonation" attack to use the effect of a Supporter?
Sure. You are not playing the Supporter, just using its effect.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)

Relevant cards

Note: All applicable cards may not be listed below.
SM12_245
SM12_245
SM12_96
SM12_96