There are 3 places within ~20 min. drive of the Refuge.
As mentioned, there's 3 Mile Canyon, and yes it's free (but pretty primitive, i.e. Porta Potties, Gravel Road, and Sagebrush).
The closest one is only ~1-1/2 miles from the Refuge Check Station right ON Patterson Ferry Road. In the warmer months it also looks (to me) a little "Primitive". Pretty much out in the open, little shade, unwatered grass, and gravel pads. It's a private-commercial venture.
In between is the NICEST one. It's ~10 min. drive from the Refuge Check Station and it's the
Boardman Riverside City Park. You just take the Boardman exit, drive toward the river, cross over the RR tracks and make a left. It has beautifully groomed green grass, lots of shade trees, 4 "Tent" sites on the lawn (only useable on the weekends, they'll have you pitch your tent on the gravel portion of an RV site, what MOST of their sites are, on weekdays), nice heated bathrooms w/showers, laundromat, and so on. I think they charge ~$15/day for Tent sites.

You can call ahead for reservations.