I am new to Alfresco web projects stuff using AVM. Wanted to understand, Is there any way to extend AVMDeployWebsiteAction class and add some more custom functionality to it.
Yes. You can "decorate" or "replace" the action if you want.
However the usual way of working with deployment is either to add an "event handler" to the deployment or to use pre and post commit deployment callbacks.
What sort of functionalility are you thinking you might need to add.