You have a project which has got some SW requirements to run (e.g.: a specific version of Apache, a version of PHP, an instance of a MySQL database and a couple of other pieces
Personally I'd suggest using Hashicorp Packer to build out your Vagrant boxes for developers, and then use it to also output a Docker or AWS or OpenStack image that you can run on your CI system. Vagrant is an awesome tool, but the overhead of VMs can be a little high for a CI system to constantly spin them up and down, especially if you want really fast feedback.
At a former client we basically booted the Vagrant machines once with the system, and then ran Docker/virtualenv builds inside of those VMs and we periodically destroyed them when there was a major upgrade or an issue with the environment not behaving correctly.
https://www.packer.io/docs/builders/openstack.html
You could try the Vagrant Plugin for Jenkins that currently supports Jenkins CD 1.532.3
You can see a demo of this plugin running at http://unethicalblogger.com/2012/03/13/vagrant-plugin-in-action.html
it is a good solution for build system, my suggestion:
Probably you can take a look at veewee, which can create vagrant box on fly.
Here is the Make CI easier with Jenkins CI and Vagrant for my guideline for this suggestion.