Other questions:
Does the campsite provide the sommercard? - You need to ask them. This card cannot be purchased anywhere - it is only available through the accommodation providers. But they are obliged to provide the Mobility Card (see post above). If you don't receive the sommercard you'll need to pay for the attractions and cable cars.
Getting from the campsite to the town for meals etc - Bruck has a few hotel restaurants and cafes but is really just a small village. To get in to Zell you'll need to take the train (see above) or a taxi, or a bus.
Supermarket - there is a big MaxiMarkt just outside of Bruck (15 mins walk?) with a Hofer (Aldi) opposite. I can't remember if there is a small shop in Bruck - there is almost certain to be a Billa or Spar in the village.
Activities - apart from the activities at the campsite, there is swimming in the lake at Zell am See (there's a lido about 10 minutes walk along the promenade from the station, just after the Grand Hotel), cable cars from the centre of Zell or from the base of Schmittenhohe (free bus from the station or bus station). Rent ebikes and take the quiet backroads around the lake (clockwise from MaxiMarkt), and on to Maria Alm or Saalfelden. There is snow at the top of Kitzsteinhorn in Kaprun, and a snow play area in the summer. There is a summer toboggan run (on rails) at Maiskogel in Kaprun.
Temperature in August - you're in the mountains, so it might be 30 deg and bright sunshine one minute, and 10 deg and thunderstorms the next, and then sunshine again an hour later. Take a light rain jacket, but otherwise August is generally hot and dry.