I have a web application that runs locally (in a local network). The application is PHP based and I am using WAMP server. The computer where WAMP is installed is always running. Is there any free software I could use to automatically backup the database at intervals and also the PHP files (at less regular intervals) to Dropbox or to some other online storage.
MySQL / MariaDB includes a nice utility called mysqldump which does the job quite handily when coupled with cron. Everybody and their brother has written a howto on scripting mysqldump from cron. Make sure to investigate the
--single-transaction option and how to store the username / password in a user options file rather than passing them as command line arguments.
I'd recommend using revision control such as Git and pushing your changes to a remote repository (like Github) rather than setting up a backup application just for your PHP files (hopefully the server is being backed up already, but if it isn't and you're responsible for backing up your own code, revision control with a remote repository will be easiest).