Do you mean the South Fork of Taylor Creek? Below Paria Point?
You can get a backcountry permit for a bivy on a route. Or, you can camp outside the park. Plenty of stuff across I-15 to the west. But you can't legally camp at your vehicle in the Kolob Canyon area. Its a short road and vehicles are very visible.
I've bivied in Cedar Canyon (you have to get about 10 miles out of town, I think), and Parowan. Neither was very good, or very close to Kolob. You should be able to easily get a bivy permit for your route, the only problem is, you have to go down to the main canyon, which sucks if you are coming from the North. There are a handful of "Ranch Exit"s along I-15 where you can pull off and bivy...but you might be trespassing. I've done it a few times, and as long as you get up early, it's not a problem.