Interesting question. In my view you do need to think about apostrophes of possession because the 'religious quests and actions' belong to the Pilgrims and the Puritans. To decide if you need one apostrophe or two (one for each word) you need to ask is there one jointly owned thing (or collective group of things) or are there two separate belongings. For example "I like John and Michael's house." and "I like both John's and Michael's houses." I feel the "religious quests and actions" of the two groups were separate things and therefore I would say "the Pilgrims' and the Puritans' religious quests and actions". However, I accept (but I don't agree!) that some people might see these quests and actions as all part of one larger thing, and so only use one apostrophe

Hi Hernandez Simon is right and the correct sentence considering also the nouns share possession rule found in " William Bradford and John Winthrop's texts" is: Explain how William Bradford and John Winthrop's texts relate to each other in terms of the legitimacy of the PILGRIMS and the PURITANS religious quests and actions”

"of the PILGRIMS and the PURITANS" does not need apostrophes they are a simple plurals. The genitive is taken care of by "of the"

John's car = apostrophe / "She said the book" = quotations