It will work the other way around, you will have to have the internship before you can get the visa. Just as in the US, there are no general visas that give you the right to work there. Also, the Thai government considers long term volunteers as workers and requires that they have a work permit. Ignore what I said about visa runs. An NGO probably isn´t going to let you work illegally, anyway.

Try the Amercan Consulate website, and look at thaivisa.com. Also, if there are any particular NGOs that appeal to you, try contacting them directly about their operations in Thailand.