Nope. They usually give you a booklet which you need to maintain. If you have the birth happen in a private hospital, they usually give it then.
But the booklet is immaterial. You can use some other card as well available from the vaccine center.
Unfortunately, there is nothing digital.
We recently lost both our children's vaccination records while shifting stuff 🙄 and had to trace back to bills, etc. to figure out where we were.
Luckily, we have been going to Apollo clinic here in Pune and they were kind enough to sift through all their bills and print out duplicates for us.
So very much need of the hour!
Whenever your child is vaccinated, they have a sticker which comes with the vaccine containing info like manufacture date, etc. which they stick to the booklet. I am assuming this would be the same in Govt centers as well?