Ok, if anyone ends up in a similar situation, you can use pkexec yum install sudo. pkexec will let you enter your password in your OS's GUI.
pkexec allows an authorized user to execute PROGRAM as another user. If
username is not specified, then the program will be executed as the
administrative super user, root.
This helped: http://askubuntu.com/questions/434525/can-sudo-be-reinstalled-after-being-removed