Recently done exactly what you are doing, why wouldn’t you be comfortable in providing a signed copy of your passport bit odd it’s your kids!

Anyway if your not there it will depend on where you are if I Thailand will need to go to amphur office and sign a permission for wife to do passport without you, if abroad need to go to Thai embassy, I must point out I emailed the UK embassy in London and they were reluctant to do it but they relented.

Really you need to be there in person, also I need more information were they born in Thailand or Abroad, because if abroad they will want birth certificates again and old passports all signed by embassy and legalisation in Thailand, luckily we were prepared.