field musicians can give the +3 to itself?

Yes — Field Musicians can give itself +3 Might.

Its "When you play me" trigger fires after the unit has already resolved into play (FAQ #2601), so by the time the trigger is placed on the Chain it is a legal "unit" for the effect. The ability has no "another" or "other" restriction, so itself is a valid choice. (Note this is a temporary +3 Might effect, not a Buff counter, so the one-Buff-per-unit limit doesn't apply.)