Can I apply for a Thai ED Visa while in Thailand on a visa-exempt status?

Can take awhile to get the paperwork together

Yes you can but hurry it takes between 15 and 21 days

It's possible but only if you start on it as soon as you arrive. Because your school needs to send copies of your passport to the Ministry of Education and wait for them to send approval for you and this can take 2-3 weeks. And then once this is received they'll prepare your paperwork to take to immigration to convert to the ED visa and you must have 15-21 days left on your existing stamp to be able to apply. So you'll likely need to apply for the 30 day tourist extension before you can apply for the ED visa.

If you don't have 4-5 weeks available on your stay then you'd need to leave and return to get more time to apply.

Ask your school

Depends how much sway your school has with the local immigration office. Your best strategy is to ask the school you're considering enrolling in and get their take on it