"Should I apply to these other departments even though I come from a Computer Science background?"
Absolutely! Degree programs vary per institution in scope and means, but the ends are often the same, after completing the program you will have a PhD in GIS. You can use this to your advantage to tailor a degree to your strengths, providing its offered at the institution(s) of your choice.
"Can a student complete his PhD program in a university department
other then the department of his undergraduate work?"
Yes, a student accepted to a GIS PhD program could complete it in a department other than Computer Science, provided the institution attending offered such a degree program.
I'm sure some schools want certain backgrounds, similar to how most seek out a certain GPA. This will vary for each university you are interested in as each university is unique and offers unique programs and opportunities.
Open data driven via GIS maps is going to rise in demand astronomically. As well as the rest of the tenets of openness shall; but in your case, I think you are tailored for a gig in open journalism, and you should focus on web development while not pursuing your PhD.
Learn web development with your GIS PhD. Your Computer Science background should make learning web development relatively effortless.
This is not a great answer in terms of a definitive answer, but your possibilities are endless: