I've made 3 trips to Assateague in the last 14 months, staying in the Maryland State Park in Loop H, the one with electric hookups. Mosquitoes weren't that much of a problem, there was enough of an ocean breeze that they weren't about that much. My understanding is that it might be different in the National Park, especially on the bay side, but maybe I've just been lucky.
As others have mentioned, ticks are plentiful, so be careful. The State Park has virtually no shade, so if you're there in unusually hot weather, even with A/C the coach might get a little warm. At least that was the case for me when it hit 98°, although temps in the mid 80's were more common. It's very difficult to get a reservation in Loop H, so you pretty much have to take what you can get.