To pull out all 40 Furies, you will need 40 Fairy Furies. Simple math, right? The majority of these will come out of story quests, but make sure to always check in with Lola before and right as you transition into a new area. You will obtain 17 from her (16 sidequests in Route #1: Goddess, as well as the components to make the 17th), 6 from Guillermo (5 for the Emblems from Shukesoo's Tower, 1 possible if in New Game+ and completing a Rank B Quest early on), and the rest from story.
Several things to note:
S Rank Rita is composed of side quests. You will obtain Broken Fury pieces throughout the story and you need the third piece from a Lola sidequest to forge. Once you've got it, you'll have to synthesize it back to normal. This will be done late in a route. Depending on the route you take, you'll either have to take it to the Cavare Sanctuary (Goddess) or the second area in Dasuhiro Plains (other routes).
A Rank Dohn is an odd one. In Goddess Route, you must complete a B Rank Quest early on to get it (which is why New Game + is often seen as the only option by then). For myself, while I was S Rank on my second playthrough, I did Vile God for my second route and actually got him as a story event item before a fight with Bernard.
DLC Furies will not work in the Between, so in order to pull all forty, you might end up having to wait until your second run. That being said, be aware of how many Furies you pull from the Vile God, as the routes you take will change on this variable alone.