Are there convenience stores inside campus? Do students need to go out of the institute to buy what they need?

You can go outside campus to shop if you prefer, but there are plenty of options available within IITM itself. 

If you're looking for grocery, stationery, items of daily use, or food you can visit:

1. Prime Mart: Located amidst the boys hostel (for specific directions, right next to Tamiraparani Hostel), Prime Mart is a well-equipped shop only a short walk away. 
2. IITM Shopping Centre: The IITM Shopping Centre offers a more mall-like experience with around 20 shops for groceries, snacks and facilities. It even has a few restaurants.
3. Himalaya Food Court: If you'd like food, the Himalaya Food Court is open till 1 am (not to mention the several messes, food court and eateries within IITM). 
4. Quark: You can visit Quark for a midnight snack or piping hot tea as it's open till 1 AM
5. Usha Cafe: Also open till 1 Am, Usha Cafe is right next to the girls hostels in the Student Facilities Centre (which has shops for groceries, snacks and facilities). 

There are many other eateries and facilities close at hand, so IITM can often feel a city within a city!

While there are plenty of options within IITM, you're not limited to campus alone. You have the freedom to venture outside and explore the nearby areas for shopping.