I wrote "Why I apply to a given doctoral program at a given university" but am unsure of the use of "at". What are the natural prepositions between "doctoral program" and "university" then?
Yes, 'at' is a suitable and commonly used preposition between 'doctoral program' and 'university'.
I am applying for the (doctoral program) at (given university).
An alternative wording that I have seen in a few places, is to use the word 'through', for example:
I am applying for the (doctoral program) through (given university).