As the weather cools, the tougher it is to get a spot. That being said, if you get there mid-morning 9-10am you should get a spot. There is also dispersed BLM camping just outside of the north entrance to Valley of Fire.

Can't say about this time of year, but if you get there early (9-10 am) as people are checking out, you'll have a better chance of finding a vacant site.

We were there a year ago. So don’t know how crowded it is now. But there are BLM sites for boondocking on both the east and west sides of the Park. They are listed on Campendium. Recommend you spend at a night at either of these and go the next morning - best time to arrive is around 8:30 - 10am. That is when most people leave.