Hi I'm a librarian, while the exact rules and procedures may vary a bit state by state, this will largely be in the hands of the Library Board. I would find out when they meet and ask to be put on the agenda. It would help if you could get a bit of a crowd together that supports your side to attend the meeting as well, and/or get as many people as you can to sign a petition for their integration.

The truth is that usually not many people show up to library board meetings, so it really doesn't take much to make a strong showing. So if you can show that there is strong support in the community for the integration of LGBTQ+ materials you would have a very good chance.

Beyond that find out their meeting schedule and attend the meeting regularly. They are open to the public and and it will help you keep abreast of what is going on at you local library and any challenges they are facing.

It could also be useful to join your local Friends of the Library group. While they don't have any official authority over the library, they tend to be a major source of donations and volunteers at the library and as such tend to be fairly influential.