If I use Mimikyu's "Impersonation" attack, do I get to use the "when you play this card" part of Tag Team Supporters or not?
Impersonation gets to use all effects of the card; you still do the entire part of the card even if you're not "playing" it (but you still have to pay any required costs).
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