The plot of The Bostonians centers on the rivalry between Olive Chance

The idiom that we need to know here is simply ‘between X and Y'.

This question complicates it by inserting appositives in between to confuse the reader.
For an appositive to work, it has to be placed right next to the word it is modifying.

‘an active feminist’ describes Olive Chancellor and ‘her charming and cynical cousin’ describes/modifies Basil Ransom.

Now, let’s scan the options:

(A) rivalry between Olive Chancellor, an active feminist, with her charming and cynical cousin, Basil Ransom

(B) rivals Olive Chancellor, an active feminist, against her charming and cynical cousin, Basil Ransom

(C) rivalry that develops between Olive Chancellor, an active feminist, and Basil Ransom, her charming and cynical cousin

(D) developing rivalry between Olive Chancellor, an active feminist, with Basil Ransom, her charming and cynical cousin

(E) active feminist, Olive Chancellor, and the rivalry with her charming and cynical cousin Basil Ransom

Eliminate A, B, D and E.

Even if you don’t know the idiom, you can use the meaning approach to eliminate the other options.

Option C is the best choice.

Hope this helps!