Are married lower enlisted authorized to have firearms in on-post housing?

You are allowed to have firearms at Fort Irwin. When you arrive go to the MP office. The desk sergeant will have some paperwork for you to fill out. It is basically a list of your weapons by make, model, manufacture, serial number. Your company commander will need to sign it. If you live in base housing you will be allowed to have your firearm in your house unless your commander has made a policy to maintain it/them in the Arms room.
There is a range on post as well. I'd recommend not going off in the desert and firing. It may not look like it but different law enforcement agencies patrol the desert for various reasons.

Know the CA laws on weapons. You're going to have to modify it to meet their ignorant requirements, and that will piss you off. Then you have to register it with the state itself. They keep a database of "assault weapons"...yeah I know. Then you need to know the rules of the installation. Don't forget to check on any of what I posted, or you could wind up with a shortened career, and a criminal record. This is California... Also being married makes no difference.
Claiming not to know the weapon laws of the state you're PCS'ing to will not hold up in court.

https://oag.ca.gov/firearms/bullet-button-assault-weapon

https://www.nraila.org/articles/20180629/california-doj-s-bullet-button-assault-weapon-registration-webpage-continually-crashing-and-preventing-gun-owners-from-complying-with-california-law

California has polices for Active duty military personnel, but don't be surprised if you have to register it as assault weapon, have some fin on the pistol grip, fix the stock. Fixing the stock is nothing more than a drill and a pop rivet. Fin can be bought. If you can avoid save yourself the hassle and don't bring thirty round mags with you if you haven't modified them for 10 rounds. Get on you tube and check for things like the patriot mag release and other California Compliant devices, you may or may not need them.