Picnic spots are easy to come by unless the elk get in there first……

It is easy to buy food and drink within the Park, but in our opinion it is often mediocre and we tend to use the in Park food emporia for provision of ice cream, cold drinks and sometimes snacks / picnic materials. If the weather is good it seems to us a shame to spend any time in a busy restaurant when there is everything in the Park to explore. On a wet and stormy afternoon, a leisurely lunch can seem like a better idea but you will not be surprised to learn it is not a unique one!

Yellowstone Picnic Areas

The Park Service provide plenty of picnic areas with benches and often long drop / composting toilets (antibacterial gel / wipes are really useful things to carry with you) and sometimes barbeque or fire areas. N.B. barbeques are called grills if you are slightly confused looking at the Park map.
You should not picnic outside a designated area and you should ensure that all rubbish is disposed of properly in the bins provided or put back in your car. This is not ridiculous bureaucracy but a measure to help reduce bear exposure to human food. The pic-a-nic baskets beloved by Yogi Bear have much basis in reality. If bears start to associate humans with food it vastly increases the chances of problem people / bear encounters and the bear may well end up being destroyed.

You will see signs to Yellowstone picnic areas from the Grand Loop Road and you are never far away from one. If you want to plan more thoroughly the Park Service provide a list and a map.
http://www.nps.gov/yell/planyourvisit/picnic.htm