Yes. When we were there in early Oct there were a number of spaces but that was 2016. Things are busier now but depending on your size I would imagine you will find something if you go in the morning.
Not sure one of these answers is correct as there is a Red Canyon site that is on recretion.gov but its in Ashley National Foreest near Dutch John. As far as recreation.gov, it doesn't list this campground. which is near Panguitch Utah.
Yes, it is first come, first serve. No reservations taken. There is a camp host with firewood and a dump station. Get site 34 if you can. It was a “double campsite” that was converted to a single and has a really spacious patio area with a built in fire ring!