There is no special visa for those wishing to intern in Argentina. Students and recent graduates wishing to come to Argentina for unpaid internships need only be eligible for a tourist visa: see the “Reconnaissance” section for more details. If it is a paid internship, you’ll have to apply for a work visa: see the section “You have a job and need to relocate”.
Changing visa status, for example from a student to a paid intern, is at the discretion of the Argentine consulate. If you need to change your visa status, you must go in person to the National Immigration Office, located at Avenida Antártida Argentina 1355 in Retiro. The office is open Monday-Friday 8:30am-1:30pm. As with anything else, be prepared for them to open late or close early. You may or may not be required to return to your home country. Be prepared to take as much paperwork as you can, including a work contract from your employer, background and criminal checks from the last 12 months from your home country, and proof of income.