Yes, everything at the resort and on the island will be in full swing in September.
The resort has a nice compliment of family activities year-round (even the winter months).
September is still warm and the ocean will be in the mid 80s during Labor Day. The summer tourism season will be winding down with few children present, but the resort will still be well occupied as well as Hilton Head Island itself.
I stayed last year from Sunday to Monday and it was excellent! I wished I could have stayed until Tuesday because they had some special activties for Labor Day on Monday evening. So there was lots to do but the island and resort were not too crowded. Weather is usually very nice that time of year also.