Do you even need a placement agency?

In other words, is there such a thing as do-it-yourself teaching?   Yes.  But it would help a lot if you have spent some time in Thailand.  Knowing the customs and a little Thai would be a major asset

Failing that, spend some time on the internet and educate yourself a little.  There is advice on teaching at this site as well as some general rules of etiquette which will help you get accepted in the culture. 

Then after you arrive in Thailand, we suggest you visit an area such as Kanchanaburi, Chiang Mai or Chiang Rai.  Spend a few days and get accustomed to the weather and the food.  Then venture out into the local community and choose a small government school in the countryside. Being well groomed, well spoken and polite will help.  Ask to speak to the Director of the school and tell them why you want to volunteer as a teacher.  Bringing along a resume and perhaps some references would help.   

If the school director is a little wary of your offer, remember, you are a stranger to them with no introduction. Schools who do accept volunteers usually have a relationship with a placement agency who screen volunteers for them.  So you have to visit several schools before you find a willing Director.  Don't take any of this personally.  It's a Director's job to be protective of his students.