Just south of Bristol there are sites nr the WSM and Burnham junctions or 5 mins from the Burnham junction a nice small site at the Watchfield Inn.
If you can spare make some time Bristol CC site is definitely worth a visit but about 15mins off route from Avonmouth junction coming from M5 North or M4 West or in on M32 if coming from M4 East.
If you just want a night stop and some pub grub I would go for the Watchfield Inn.