Yes, there are grills at SSR.
In the Grandstand section, there is a picnic area next to the water play area at the pool. It has two grills and two picnic tables under an enclosure. These are the ones we've used in the past. Our room overlooked them, and I can tell you they get used almost every night.
There are also grills by the quiet pool at Congress Park. I think there are tables here, too.
There used to be grills near the Paddock quiet pool, but I don't know if they're still there after the refurbishment and creation of the new restaurant.
Also, every tree house in the THV section of SSR has its own grill. Part of the whole "Glamping" experience for us was to cook out often during our stay there.
You can check out a "grill kit" with standard cook-out utensils from the community hall. The tree houses have grill utensils as part of their standard kitchen inventory. Be sure to pack your own matches, though. We had a heck of a time finding a way to get it lit.