Our logistics department handles that.
I am required to do an inventory of my first out, airway and ALS airway bags once a month and to replace anything expired. Stock-wise it's always fine because I either work out of supplies on the truck or restock my bag with whatever was used right after the call. I don't rely on my partner to do it, even though she always does cause she's awesome. In the end they're my bags, my responsibility.
I believe they use a program called Ambutrak for inventory. Well at least to keep track of laptops, radios, GPSs, monitors and pumps (for CCPs) when we check gear in and out. Pretty sure it does our inventory too though.
The company I work for uses "speed loading" to stock rigs. As long as the VST who put the containers together didn't have their head up their *** it works fantastically. Only issue I have is every VST puts stuff in the containers differently. Every container has the same stuff in the same spot on every ambulance. The organization of the supplies in the container is what tends to vary.