1- Delete the relevant section from the .gitmodules file.  The section would look similar to:
[submodule "vendor"]
	path = vendor
	url = git://github.com/some-user/some-repo.git
2- Stage the .gitmodules changes via command line using:git add .gitmodules
3- Delete the relevant section from .git/config, which will look like:
[submodule "vendor"]
	url = git://github.com/some-user/some-repo.git
4- Run git rm --cached path/to/submodule .  Don't include a trailing slash -- that will lead to an error.
5- Run rm -rf .git/modules/submodule_name
6- Commit the change:
7- Delete the now untracked submodule files rm -rf path/to/submodule