We camp at Pontiac Lake state park regularly (couple times a year). Not "close" to the zoo, still about 45 minute drive, but about 15 minutes west of Pontiac. A large campground and rarely sold out, even on holiday weekends.
Proud Lake SP is also an option. It's probably easier to get to the Zoo from there as it's just a couple miles off I-96, so practically all freeway to the zoo, probably 30 minute drive. It's s/w of Pontiac, near Wixom. I've camped here as well, but it's been a few years since the last time.