Is it appropriate to request that the department hire a female faculty member?
the new faculty should be female in order to help address the wildly disproportionate gender ratio in our current faculty.


This is not a good reason. Gender imbalance is fought by educating everybody (males, females and any possible group) equally and hiring the best people, regardless of their gender, not choosing people by gender; that's sexism.


  The candidate that we support is highly qualified in terms of research and teaching. Moreover, among all the candidates who have visited, they are the only one who have any experience with diversity initiatives and getting underrepresented groups into stem. This is very important to many of us, especially since the department is planning to add an undergraduate program in the near future.


These are very good reasons to hire someone, and reasons to be proud being hired for. If I were her, I'd find disrespectful being hired first because I'm a woman and second because of these good reasons. 


  We are planning to write an open letter to the department in support of this candidate, citing her qualifications as someone who has worked to make science a more inclusive place for women and other underrepresented minorities.


Again, these may be good points to write in the open letter, but the fact that she's a woman shouldn't matter. She's a great candidate no matter what her gender is. 

I find this logic of hiring women because they are women very sexist, towards both men and women. Towards men because they are at a disadvantage, towards women because you're treating them like kids, giving them a preferential route they don't need. Women can clearly be good enough to be hired just for their skills and not for their gender. 

Fight for her to be hired if you think she's the best choice. Write the letter and explain why she's the best choice, that she's better than the other candidates because she is better, not because she's a lady.