Internships have a special work permit process. It is completely separate from becoming eligible to work as a language student after a year of study.
The company applies to the Ministry of Economic Affairs (in most cases, depends in the industry) for an internship permit. If the Ministry approves the application they issue a letter. The intern takes the letter to a Taiwan rep office overseas and obtains a visitor visa for the purpose of an internship. If you are already in Taiwan on an ARC or a Visitor Visa for the purpose of language study, you can probably get away with just taking up the internship without exiting and reentering on the new visa. This is a bit of a grey area because the authorities could think that you have violated your purpose of entry, so the really cautious thing would be to exit Taiwan and return on the proper visitor visa. You can continue your language studies with no problem.
Here is a link that explains the process. You can show it to your potential employer and confirm that this is what they are doing.
boca.gov.tw/content.asp?CuItem=2065
The MOEA (and other agencies) approve internship applications routinely these days although they prefer that the internship be unpaid. If it is paid, I would recommend not exceeding NT$200/hour.