Can my child run to first base on a dropped third strike?

Yes, in Majors. The uncaught third strike rule applies from the Major Division up (it doesn't apply in Minors or Tee Ball).

When the catcher doesn't cleanly catch strike three, the batter can run to first if:

• first base is empty, or
• there are two outs (even if first base is occupied).

The defense puts the batter out by tagging them or first base before they get there. With first base occupied and fewer than two outs, the batter is simply out.