After reading the link from @aloctavodia and consulting the massmutual/mmviz-python GitHub repo, this is what I came up with.
setup.py
from setuptools import setup
from setuptools.command.install import install
import os
import shutil
import atexit
import matplotlib
def install_mplstyle():
    stylefile = "mystyle.mplstyle"
    mpl_stylelib_dir = os.path.join(matplotlib.get_configdir() ,"stylelib")
    if not os.path.exists(mpl_stylelib_dir):
        os.makedirs(mpl_stylelib_dir)
    print("Installing style into", mpl_stylelib_dir)
    shutil.copy(
        os.path.join(os.path.dirname(__file__), stylefile),
        os.path.join(mpl_stylelib_dir, stylefile))
class PostInstallMoveFile(install):
    def __init__(self, *args, **kwargs):
        super().__init__(*args, **kwargs)
        atexit.register(install_mplstyle)
setup(
    name='my-style',
    version='0.1.0',
    py_modules=['my_style'],
    install_requires=[
        'matplotlib',
    ],
    cmdclass={
        'install': PostInstallMoveFile,
    }
)
Into my_style.py I just put a basic example script. Now, my users can install this style with pip install git+https://github.com/me/my-style!