Penultimate part of Shakespearean play is beginning to move into vigorous campaigning (8)
'vigorous campaigning' is the definition.
'penultimate part of shakespearean play is beginning to move' is the wordplay.
'penultimate part of shakespearean play' becomes 'activ' (fourth of five acts in a Shakespeare play).
'beginning to' suggests taking the first letters.
The first letter of 'move' is 'm'.
'into' is the link.
(Other definitions for activism that I've seen before include "Political efforts", "Policy of vigorous work as in politics".)