I will be hosting a Django application on hostgator.com and was wondering how can I install django apps and python packages without using pip, since I won't have access to the root account.
Shared hosting accounts rarely, if ever, allow packages to be installed on their server as it will affect all sites on the server and they can't allow that. Plus for the price you're paying they wouldn't want you to. This usually requires a VPS or dedicated server.
Depending on Hostgator you may be able to use
virtual-python.py to create a local copy of python in your home directory and install packages within that copy.
Python packages are no more than just directories and files. You should upload them and then point to them using your Python code. Check whether or not your web host supports virtual environments.