I would like to setup a workflow that enables pushing local changes to Bitbucket or Github (but ideally the former) that then updates a live site hosted on my lamp VPS server. Is this possible? I remember working some time ago with a team that had a similar setup but with Github.
There is an article by Joe Maller http://joemaller.com/990/a-web-focused-git-workflow/ that describes a similar approach where the hub is another directory on the same webserver.
That kind of post-push action is managed:
In both cases, those hooks generate a POST hook which passes a payload to your third-party web application through its API.
Your third-party application is responsible for receiving and transforming the payload.
That means your lamp VPS server needs to have a listener in place able to interpret said payload.
you can see an example of a script managing a deployment in this article, by doing a git pull:
// Make sure we're in the right directory
exec('cd '.$this->_directory, $output);
$this->log('Changing working directory... '.implode(' ', $output));
// Discard any changes to tracked files since our last deploy
exec('git reset --hard HEAD', $output);
$this->log('Reseting repository... '.implode(' ', $output));
// Update the local repository
exec('git pull '.$this->_remote.' '.$this->_branch, $output);
$this->log('Pulling in changes... '.implode(' ', $output));