Yes. Go to the "official website" link on the Campendium page and call the number. There are only 12 sites, so you should make a reservation. If you show up without a reservation, look at the chart on the office door at the park entrance, choose a site that's vacant for the period you want, and write yourself in on the chart.

Per the document accessible from the official website, you can make reservations by calling.

It has been a couple years ago but I think if you can find Bingham County and check https://www.co.bingham.id.us/parks_recreation/north_bingham_park.html