I ride my kart around on my neighborhood streets, the neighbors love it, and I take safety precautions. From a parental standpoint however I can understand why that isn't really an option for your kids. Yeah, look for empty lots like you already mentioned, developing neighborhoods when they're not bust with construction workers, and also try church parking lots. If you go to church, check with the pastor or whoever's in charge, my bishop's totally cool with it, he actually has a kart for his boys lol.
My experience is that if you take safety precautions, share the road, and be considerate, people actually love it.