"does iOS and Android has a built-in support for SQLite or I have to deploy/install something for this as well?"
For a list of ways to store data see my reply here:
"I want to know if I can make an existing SQLite DB the part of Worklight deployment package for Android and iOS?"
Not sure, I would build a simple REST service (or similar) and use Worklight Adapters to add data to your mobile Worklight Applications.
See the articles on Adapters here:
Alternatively it's easy to work with SQLite on Android and iOS, you could build a Cordova plugin and write native code for both platforms.
See the articles on Cordova here: