The sentence, as written, is grammatically correct but logically weak and stylistically awkward. In order to improve it, we have to look at the relations between the ideas--to "unpack" the sentence. Here is a breakdown of the relations:
a. The housing affordability gap is widening. [BECAUSE OF THIS,] middle-income families are especially hard-hit.
b. [THIS IS BECAUSE] they no longer qualify to buy homes [and AS A RESULT must continue to pay rent.]
c. [BECAUSE they are paying rent instead of paying for a home,] and BECAUSE rental rates are rising, they are forced to use far more than the standard 25 percent of their incomes for housing.
d. FURTHERMORE, [BY PAYING RENT INSTEAD OF INVESTING IN A HOME,] they are left with no equity as well as no tax write-offs to offset the expenditures.
Obviously, not all these meaning relations need to be made explicit: the reader can supply some of the connections him/herself.
Still, the sentence has so many different ideas that, for the sake of clarity, it's better to break it down into two sentences. It's also better to use sentence and paragraph structure to do some of the connecting of ideas. Here is one possible version. Numbers in parentheses mark the notes below:
As the housing affordability gap is widening, middle-income families are especially hard-hit. (1) These [middle-income] families, (2) who can no longer qualify to buy homes, (3) are not only forced by rising rental rates to use far more than the standard 25 percent of their incomes for housing, (3 )but are also left with no equity or tax write-offs to offset the expenditures.
(1) At the beginning of this next statement, the reader already expects an explanation from the writer for having made the prior statement, and therefore a logical connector is not necessary.
(2) Often a nonrestrictive relative clause can provide the reason for the following statement, making a connector unnecessary. (In this case, the relative clause has to be nonrestrictive, since the writer has included ALL middle-income families in the opening statement. Such a statement seems too general to me, but that's the way it's written.)
(3) 'Not only...but" introduce the two consequences of the families' inability to buy a home.
This is one way to fix the sentence, but obviously not the only way. Would readers like to try other versions? Please feel free.