My son was born in Chiang Mai, has a Thai birth certificate translated to English. 
An embassy is asking to legalize it by MOFA. 
Is it possible to do it in CM? Couldn’t find the office location online. 
Second, we are not in Thailand at the moment but the certificate is there. What would be the best and cheapest way to have that done?

The MOFA has a postal service.

http://m.mfa.go.th/main/th/services/1303/24762-Procedure-of-Document-Authentication-by-Mail.html

Do you mean legalize the Thai birth certificate? 

If the B.C is written in both Thai and English (Which they usually are), it shouldn't need to be translated 

17 minutes ago, Neeranam said:

Do you mean legalize the Thai birth certificate? 

Yes

15 minutes ago, Mac Mickmanus said:

If the B.C is written in both Thai and English (Which they usually are), it shouldn't need to be translated 

 not translated, legalized power embassy request. 

The Department of Consular Affairs office for legalizations and certification is located at the Thai passport office in CM.

Map and contact info is here in Thai (google translate works on it)

 https://consular.mfa.go.th/th/publicservice/สถานที่ให้บริการรับรองนิติกรณ์เอกสาร

The can do the certification of a Thai birth certificate. Info here in Thai. https://consular.mfa.go.th/th/publicservice/การรับรองเอกสารที่ออกโดยหน่วยงานต่างๆ

On 7/12/2021 at 1:56 PM, Mac Mickmanus said:

If the B.C is written in both Thai and English (Which they usually are), it shouldn't need to be translated 

My son, born here in 2002, has a birth certificate, which is only in Thai. We got it 'certifide' at the Amphoe and it was translated and the translation 'certified' by a local law office.