Do children under 6 need to apply for a 30-day extension on a tourist visa in Thailand?

Ideally you should get an extension for the kids. If you don't they will be on overstay and their passport marked appropriately. There are no repercussions as regards Thailand but there may be if you intend to apply for visas to visit other countries

In a addition to what

@Stuart ********

has said, if you want your kids to attend school in Thailand they may need to be on a visa/extension.

Kids under the age of 15 cannot be fined for overstay. On exit they will get an overstay stamp in their passport but it would mean no repercussions for re visiting Thailand. Other countries may not hold similar views. There could be potential issues applying for visas for those countries. Whether you choose to apply for extensions for your children is on you.

You won't find the information online because it doesn't work the way you think it does.

Yes everyone needs a visa or extension, so your child needs one.

But there is no overstay fine for children under 15 and you cannot be banned from Thailand for overstay while you're under 18.

So if you do not get an extension for your child, when you go to leave at the airport they'll receive an overstay stamp in their passport but will not fined.