How can I find a biking club that has a group for riding with kids in a trailer?
Besides talking to individuals in your local bike shops, you can look for 'recreational' or 'touring' clubs. A Google Search for Bicycle Touring Clubs turns up clubs like our club in Utah, Bonneville Cycling Club (formerly the Bonneville Bicycle Touring Club).

Larger recreational or touring clubs will usually rate their group rides by a couple of measures like distance and average speed, and will sometimes classify rides as drop/no drop (meaning that in a 'no drop', the group slows to the speed of the slowest rider).

Pick the slower and shorter rides to begin with, and be prepared to be fully self-supported.  I would welcome riders with a trailer or trail-a-bike on most of my recreational rides, but it can be frustrating if those riders really can't hang with the regulars.

Finally - try to preview the group's route if possible.  Most recreational group rides stick to quieter roads with wide shoulders, but you need to be aware that with the trailer you will be much harder to pass.