How to clone my own Laravel project considering vendor ignored

I'm looking for help on dealing with the whole setup process of a Laravel project. Currently this is my walkthrough list:

• Install Virtual Box and Vagrant.
• Run vagrant box add laravel/homestead
• Run git clone http://ift.tt/SBuys2 Homestead on your favorite folder, using your favorite Bash (in my case I'm using Git BASH because I'm on Windows 10)
• Setup your SSH Keys with ssh-keygen -t rsa -C "email@domain.blah"
• Setup SSH connector file[¹].
• Setup your Homestead.yaml as you wish.
• Run init.sh / init.bat.
• SSH into the Vagrant.
• run composer global require "laravel/installer"
• run laravel new project
• Go back to the host machine, into the project folder and run git init, git add ., git commit -m "clean project"
• Push the project with git remote add origin http://ift.tt/1NXPB0q and git push -u origin --all

Now I have a brand new Laravel project hook up to Git for versioning. My problem is that Laravel ignores /vendor by default. Considering this fact, I want to clone my project in another computer because I have 2 computers to work and/or a co-worker wants to clone the same project so we can work on it together.

What would be the proper walkthrough to clone the project and have Laravel work on another machine? Do I have to add /vendor to the repository and push? Should I add the homestead box into the repository? If so, how?

Thanks in advance.

[¹]
Host homestead
HostName 127.0.0.1
User vagrant
Port 2222

via Chebli Mohamed