While other uses are possible, the two main uses of Vagrant are to instantiate virtual machines with a defined configuration for either testing or for a particular ensemble of development tools, reproducibly.
The virtual machines created with vagrant are not particularly different from VMs created using other methods; they may be a bit less secure because of some considerations that went into vagrant's conventions; they are reproducible to the extent that every version of every kernel module, program and library is initially fixed in the "box" file that the VM is created from.
But for a coworker to take advantage of Vagrant's features, they would need a vagrant setup of their own. Given a copy of your vagrantfile, they could then produce a replica of your vagrant machine, but it's considerably more involved than a URL or click.
The problem you are trying to solve does not look like a particularly good fit.
I read your question as wanting to know how to reproduce a working development environment where you may not have started out with strict control over all of the versions of all of the modules, libraries, or other development tools. You might be able to use a Vagrant-related tool like Packer to freeze a particular existing environment for purposes of reproducing it elsewhere, but it does not truly solve the underlying problem of gaining control over the library dependencies, etc.
With most VM hypervisors or tools like docker, there are ways to snapshot any existing workspace so that it could be duplicated and shared with someone else. And there are any number of configuration and provisioning tools like Ansible, Puppet, or Chef that allow you to more closely control and document the building of your development environment so the configuration can be more controlled (and also regenerated).