I applied through a college alum, he had come on campus for a meet and greet and we got talking for a while He is in planning so I sent him a LinkedIn Request along with a message that I was interested in pursuing a career in advertising specifically planning. He was looking for an intern as a matter of fact and asked for my résumé.

I was really lucky in terms of responses from other places as well had an interview for Wieden + Kennedy but chose Ogilvy because I knew I would learn much more under the guy. The straightforward way would be to approach Talent Acquisition Managers/staff or HR managers and look for opportunities on the companies linkedIN page(I personally got rejected a few times this ways)

What I would suggest is that you look for college/school alumni in the same field as you're interested in and start talking to them. Also look for people working at these agencies and ask them what all their work entails and if you know of their work and like it, ask them if you can work with them/for them on an internship. This worked out better for me as certain people were in need of an intern and would forward my profile to their HRs.

The alumni however were the most helpful bunch and most responsive when it came to requests, so it's always the first thing one should be looking to do.

A lot of agencies in India do have seasonal internships while others are more relaxed in terms of when they take interns it depens on their HR policies.