You can go to Wahweap Campground close by and take a shower for a small amount.

Apparently not. This from the nps.gov (https://www.nps.gov/glca/planyourvisit/camping.htm): Lone Rock Beach Primitive Camping Primitive camping permitted on a sandy beach or in dunes. No designated campsites. Open fires must be within a four-foot-square area. Quiet time 10 pm-6 am. Four micro-flush toilets, 6 vault toilets and 1 wheelchair accessible comfort station. Outdoor cold shower. Off-road vehicle area. Dump station. Potable water (seasonal). Day use area. No launch ramp. No reservations. $14 per vehicle per night. Camping fee does not include entrance fee, and entrance fee does not include camping.