For the type of reference letters you mention, I think they normally want the letter submitted directly to the program (or employer) for the internship.

Without knowing the specifics of the program in NZ, it sounds like perhaps you should fax the office organizer and ask her to forward the message for you. Be sure to explain to the group leader the type of program you are applying for and why you would like to do it (what you think it will do for you, how you are particularly suited for their needs, and how it relates to what you did in NZ). That gives her a little bit to work with when writing the recommendation for you. Don't tell her what to say, but do make clear why you are interested in the position.

You may want to give her an "out" - just in case she is too busy or can't write a reference for you for any reason (some people just hate writing these things). That gives you the chance to find someone else (like maybe the office organizer).
Cheers,
Bev