You need to visit your nearest LIC Branch to get to know about the person/agent who takes care of LIC Cards, LIC branches don't deal with LIC cards directly, however they have the contact details of the agents who can process the LIC card request.
You need to reach out to the agent recommended by the LIC branch, there is an offline form which the agent will fill on your behalf, he will also be needing following documents to process your application.
1. Your last three months salary slips (if you are salaried) or ITR from last 2 years (if you are an entrepreneur)
2. Bank statement of last three months
3. Your passport size photograph
4. Copy of Address Proof
5. Copy of Identity Proof
6. Copy of your LIC policy
Agent will submit your application and then the wait starts, if everything goes fine your application would be approved in 15-20 days and you will receive the card in a month.