The process is as follows:
- You arrive to Switzerland on a tourist visa (that is, unless you already are in Switzerland or have any other legal means of getting into Switzerland).
- On the first day of your employment / internship, and upon signing the contract, you are given the forms to fill out for the carte. You will also attach to these a photocopy of the document authorizing you to stay in Switzerland / Schengen Area.
- The organization then sends the paperwork off to DFAE (Department federal des affaires etrangeres - or so it says on my carte, anyway!).
- You will receive the carte in about two weeks.
You will have the H carte, which does
not grant you immunity from prosecution or access to the Swiss labour market. The first provision changes depending on the seniority of the carte holder.
Note that exceptions to expedite the above process are possible, but are unlikely to apply to interns.
Good luck!