Can my son apply for a 60-day visa extension online or does he need to visit the immigration office in Chonburi?

You can apply for an e-extension online for a 30-day tourism extension at the pattaya office paying for extra fee addtional to the official fee of 1900 baht. But you need to go to the pattaya immigration office in person to get the e-extension sticker after application got approved within 1 to 7 business days for processing (time depends on the fee).

Or go to the local immigration office in person directly to apply fo an extension within the last week of the current stamp.

Either way you need to go to the immigration office.