Totally depends. You can sign a contract with a realtor to find you an apartment. There is usually a fee. Usually one month. Yes the people renting will also have a contract with the realtor and will pay. What I would do is: go into a real estate office. Ask if they have a print out of their available rental properties. Tell them you're moving off base, don't waste the opportunity to mention that! They might have a program or a discount or something to help you out. If they are a member of your community they would be dumb a**es not to try and help out members of their own community who sacrifice and risk it all for them, ya' understand?
If you at any point feel they might be shady, leave. People still in real estate need you more than you need them! Don't sign anything that day!!! Make a rule you must sleep overnight on it or something like that. These real estate folks are going to show you their clients houses for rent first. They make money to find renters for their clients period. Keep your power.
Don't forget to check with the base. You never know what services might be there for you. Like a lawyer to read over any contracts you need to sign. I don't know for sure but that seems like something they could help with.
Make sure you do research on your own: what are the schools like? How close is the supermarket? How busy is it at the times you need to drive? Do you ever want a dog? Do you want a tree house? Does the block have a lot of vacancies/ why if yes? How much money goes to the schools? How's the local infrastructure (are the roads a mess/not enough stop lights/too many punishment lights(traffic light for what appears to be no reason and you sit and wait not for cars but for the light to change), what are the cops like? Do you want to live here for a year or years? Ect.
Since you are in the service what kind of length of contract could they make available. Most people are not a**hats and are willing to help you out. You just gotta ask! Good luck!