working on a dependent visa

My wife entered Japan on a dependent visa. I'm working full time and have the work visa. She has been offered part-time work. What does she need to do to be able to work part-time with her visa status? Is there a form to fill out and if so with whom?

She needs permission from the immigration bureau to engage in part time work.

I think the following page shows the material needed:
http://www.immi-moj.go.jp/english/tetuduki/kanri/shyoru...

But I recommend to consult with the immigration bureau directly.

Is the immigration bureau the same as the kuyakusho where we get our gaikokujin registration cards?

acase,

No, we would refer to "kuyakusho" as "city (ward) hall." Immigration bureau is "nyukoku kanrikyoku," and their locations are listed here:
http://www.immi-moj.go.jp/english/index.html