Yes, you can open a trading and demat account with Zerodha even if you already have existing accounts with other brokers. Indian regulatory authorities use your PAN card to identify all your investments, regardless of how many demat accounts you hold.
You can maintain multiple trading and demat accounts with different brokers simultaneously. You do not need to close your existing accounts before opening a Zerodha account.
How to open your Zerodha account
You can open your Zerodha account online by visiting signup.zerodha.com and following the simple on-screen instructions.