I'm looking at driving to Crater Lake from Portland. I don't have a tent, so I'm wondering if I can sleep in my car overnight there?
You cannot overnight inside of most National Parks in your car, but you can in the National Forests than surround them.
Crater Lake, for example, is surrounded by Umpquah National Forest. You can boondocks there for free. This screen shot from the "public lands" app for iPhone shows it:
There are two "dispersed camping" areas- Twin Lakes and a shelter.
Typically you just need to be 250' away from a road. Your best bet is to sleep in a tent, but often you can find a Forest road that you can park off of. Your absolute best bet is to find the forestry office or visitor center and ask there where the best place to sleep would be. Absent that, just find some place out of sight and you'll be fine.
If where you are planning to stay is a campground, technically the answer is no. What I have done in the past, however, is take a water-resistant blanket/poncho, tie it to my car, or a couple poles, then sleep in my car. When asked by officials, I was ready with my answer: I WAS sleeping under my lean to, but I got cold/wet/swarmed with ants.
According to this page Lost Creek Campground is TENTS ONLY, and Mazuma campground is RV down to tents. I don't believe any National Parks/National Camp Grounds allow indigence camping, however.