How do I request permission to live off post?

You will need to talk to your unit command once you get boots on base. They are the ones that bless off on their Soldiers living off base. BUT.....Do not be surprised if your request is denied.

Carrying animals from post to post is a rough issue.
As a dependent kid, I always had to give up my pets before a family transfer as we never knew where we would have to live when we arrived and when housing would be available. That was a long time ago, but such uncertainties still exist.
That limbo time between arriving at your new base and finding housing that will allow pets is tough on a family and double tough if you are single. Long term boarding during that time is also a major expense. A possible resource would be the base vet, if there is one. Not all bases have them.
I had to turn my cat over to my parents for 5 years while I was on active duty and then do it again when my husband was transferred to Gitmo Cuba. Yes, you can take pets overseas, but you should look into the requirements, possible confinement and quarantine rules. I found it was not worth putting my pet through that kind of long term stress.
Good luck to you.