I realize that some of the things vagrant can do seem almost magical, there is nothing magical about backing up a VM. It's almost certainly not wise to try to use vagrant package to perform a backup.
If you're taking advantage of vagrant, then what you should be aiming for is the ability to recreate the VM but perhaps in a different location or even with a different provider. Careful separation is needed between system and application software -- which vagrant can install in a consistent fashion -- and user configurations and data -- which have to be handled separately, unless they follow a pattern you can develop vagrant recipes for. One example is keeping an SQL database attached to an application on a separately mounted volume.
Vagrant has no facilities in itself to aid in making backups. It does not take snapshots or communicate with databases and file systems to establish quiescent backup points. The operating system of the guest VM and its VM host may offer such facilities and you would need to use those to get usable backups of a running system. It would most likely all be a separate apparatus from vagrant. If you are serious about no-downtime backups that can actually be recovered, I suggest that you may need to invest in a commercial software solution and the choices vary depending on what OS your guest and host are running, and on what hypervisor. If you were thinking vagrant package was a solution, I'd be remiss in letting you go on thinking that you could just roll your own backup system without additional help.