The reason for the access denied
error is that mysql by default limits access to databases to the local machine. It is possible to open up for remote connections as described here, but it wouldn't be a good idea for production environments.
A better way is to add aliases for the commands you want as described here.
For doing calls to artisan etc in code, Laravel has a built in way of running commands on remote servers. First add connection information to the app/config/remote.php file. For vagrant it should look something like this:
'connections' => array(
'production' => array(
'host' => 'localhost',
'username' => 'vagrant',
'password' => 'vagrant',
'key' => '',
'keyphrase' => '',
'root' => '/vagrant',
),
),
Then execute artisan migrate like this:
SSH::run(array('cd /vagrant', 'php artisan migrate'));
You can do more advanced things as well. Here is the documentation.