I believe most airlines require the purchase of a separate ticket to earn miles. Lap children just don't cut it.
Here's another way to earn your kids miles...enroll in a program like IDine which gives Delta Skymiles. Route the miles to your kids account. That $30 you spend at the pizza parlor = 300 Skymiles. I haven't set it up yet but unless they've got some restriction, I don't see why it wouldn't work.