My US citizen father was never married to my mother, can he still sponsor me?
In order for your father to file for you for a green card, you have to be legitimated and those laws depend on where you were born or the country of birth.
Some countries will legitimate you just by having your name on the birth certificate. Other countries only allow legitimation if your parents are married. If you are from a country where you can only legitimate by being married, then unfortunately they probably will not be able to sponsor you.
If a father who had you out of wedlock wants to sponsor you from a country with easier laws, they have to demonstrate that you had a father/child relationship prior to your eighteenth birthday.
If you have legitimization and evidence of a father/child relationship, than yes, your father can sponsor you for a green card regardless of whether your parents were married.