Nearly every town I have lived in of size has had a business that sold glass for just this sort of thing. When I moved into this house, there were three different windows that were broken. I just called them up and they came out to measure them. A few days later, they were back and installed the glass on all three windows on site.
Replacing the door would most likely be more expensive and more time consuming and not give you any improvement over what you have so there is hard to see an upside to going the door replacement route.