Can I apply for a Non-O retirement visa online while on a 60-day visa exemption in Thailand?

You need to prove you have 800k baht or equivalent in order to apply for 90days nonO since u don't have a thai acct u can't do it in thailand (unless agent help you) if you do it in other countries does the thai embassy in that particular country accept your prove of funds?

Border, not boarder

Easiest way is to take a 2wk vacation to Vietnam and use evisa from there. Then come get your bank account, etc ...

You can apply to convert your visa exempt entry into a Non-O visa at your local immigration office providing you meet the requirements. Generally you need to have a minimum of 15 days left on your current stamp (21 days in some places) and to have a minimum of 800k baht in a Thai bank account in your name, many offices require this to show as having been transferred from overseas

FYI

You can convert visa exempt entry to non o retirement while in thailand

No, you cannot apply online using evisa while in Thailand. You must leave Thailand, go to a country where the Thai embassy accepts visa applications from non-residents, and apply through the Thai embassy where you PHYSICALLY are. The application specifically asks for your CURRENT PHYSICAL LOCATION. One of the questions in the application also asks for proof of your current location. And the application states you must remain in that country until your visa application is processed.