Can I obtain a Thai bike driver's license while on visa exempt entry?

You don't need a Thai licence if you're a tourist. Your home country licence will suffice

Edited: normally not possible (the regular way) on an exemption on Koh Samui (as mentioned by the OP)

Perfectly possible, many people have do it... But if you are not planning on living here I would not. You need a resident certificate though. Which you can get at your local immigration (requirements vary greatly by office) or through your counties embassy (usually just done as affidavit).

Someplace have agents to thay go with for the license to translate and help you through the process. Not sure where you talking though.