The sentence is as follows:
The association "Association Name" requirements.
Where should I put the possessive apostrophe?
The association "Association Name"'s requirements.
The association's "Association Name" requirements.
No matter where you put the apostrophe in that example it's going to look strange to some people—regardless of whether or not it's considered correct.
In this case, I think the apostrophe should go after the closing quotation mark—but the combination of those two punctuation marks ("'s) doesn't produce a clear presentation. And adding a space as you might for embedded quotes (" 's) doesn't work well for the possessive.
The best course is to rephrase the sentence itself in order to bypass the need for an apostrophe.
The requirements of the association "Association Name" are . . .
The possessive apostrophe comes with the immediate preceding noun. So here, the apostrophe would be added with the proper noun.
Viewing a similar but easier example:
"This is my sister Jane's book." is correct
"This is my sister's Jane book."