Hi Lynn! The Wine and Dine half marathon is my favorite Disney race! Your husband will love it!
You can certainly bring your children to the post-race party. There is nothing "wild" about it. There will be alcohol available at different food vendors, but most of them sell it all the time anyway. It will be starting very late though, so you may take that into consideration with children so young.
If you didn't want to bring the kids, you can always use the
Kids Nite Out babysitting service. These are screened and licensed caregivers that can babysit in your room at your resort. I use them for my two boys when my husband and I both run a race.
Since the Art of Animation is not a host resort for that weekend, you will have to arrange for a taxi to take you to and from the race. The Disney buses only run to and from host resorts. I have used a cab before and as long as you get the road closure information from your resort front desk, you shouldn't have any trouble!
I hope this helps! Please let me know if you have any other questions!!