adelia is right. Contact local pit bull organizations, or even national ones, such as Pit Bull Rescue Central, and they may be able to help. Contact shelters and humane societies in the area that you are looking to move to. If they take in pits, they may be able to help as they will know what pit owners have to deal with.
Get a list of pet friendly places that you are interested in and call the leasing office directly. Just ask what their pet friendly policy is and if they have breed restrictions. If they ask you what breed you have, don’t answer. Just tell them you would like to know what the restrictions are, if any. By telling them what you have, that person could be biased and tell you pits are on the list, even if they aren’t. You aren’t being dishonest; you just aren’t offering up info at that time.
There are many rental homes out there that are pet-friendly. Don’t forget to check those too! Because of the housing market, many people who can’t sell their home are deciding to rent it out for sometimes less than rent at an apartment! Many homeowners will have looser requirements for pets than large apartment communities, but you just have to ask.
All you have to do is ask. I am glad to know that you are looking for pit-friendly housing and wanting to be honest about it. People who lie or hide the dogs if moving into a non pit-friendly location just make it harder for everyone else.
Best of luck to you and the family! 🙂