Yes, you certainly can go to Sentinel Dome yourself. One of the tours charges $118 for adults and $59 for kids, but that includes the drive there and a meal. That particular tour started from Oakhurst and Tenaya Lodge, but there may be ones that start from Yosemite Valley. The park concessionaire, Delaware North, offers a stargazing tour from Yosemite Valley to Glacier Point and back, but that one does not include Sentinel Dome.
You can drive yourself there if you want to. It's about a 1-hour drive from Yosemite Valley to the parking lot for Sentinel Dome. It is about a 1-hour hike from the parking lot to the top of Sentinel Dome, then a 1-hour hike back. But it is not a major climb; the elevation gain is only 300 feet. You will still have to make that hike even if you take a paid tour. There is no way to drive to the top of Sentinel Dome, in contrast to Glacier Point and Washburn Point, which can be driven to.
The nights of the full moon this summer are June 15, July 14, August 13 and September 12. As an added bonus, August 12 and 13 are the nights of the Perseid Meteor Shower, and Sentinel Dome would be an excellent location for viewing it. However, you would have to stay up past midnight to see it at its peak of approximately 1 meteor per minute. Since you are on vacation, I assume it is OK for your kids to stay up past midnight. Also, because August 13 is a full moon, the extra light would reduce the visibility of the faint meteors, but you would still see the brighter ones.
As the top of Sentinel Dome has an elevation of 8,122 feet (2,475 m) above sea level, it will be chilly at the top at night, even if the day was hot. So you should be prepared for this by bringing a blanket and pillow for each person, and a flashlight for each person for use on the trail on the way back to your car. I think it would be an excellent idea to pack a light picnic supper, drive to Sentinel Dome, make the hike to the top of the dome, watch the sunset and orange alpenglow on Half Dome and the peaks of the park as the valley below descends into deep indigo, then eat your picnic supper and see the full moon illuminate the granite domes and walls, and stay up there for some stargazing and possibly the meteor shower if you are there then. This could be one of the major highlights of your vacation.
Remember that the park does not allow you to keep food in the car after dark, so any picnic supplies should be taken with you and fully consumed, and pack out your trash and dispose of it according to park regulations.