Start contacting the council of labour relations. They will show you the way. I think it is actually processed by MOEA.
We recently brought an intern aboard, and the process was not especially tough, the most important is the documentation that you will be a student while you are an intern. They did not ask for certifications of documents or any of that crap, it went fast and smooth.
If you are here on the right type of visa, you do not need to leave the country, our intern was not, so he had to go to Hong Kong.