As a student, is it appropriate to use my university's letterhead for a letter I write, to give it extra weight? A program I participated in for six years, while I was in junior high and high school has just had a huge chunk of their funding removed by the school. I credit this program with the majority of my success in college and my general interest and capabilities in science, and a lot of my maturity (which was sorely needed in junior high...).

The program has a new coach (who I worked with, but has just now taken over) who is not a confrontational person. I would like to write a letter to the school expressing my support for the program, and would encourage other alumni of the program to as well, many of whom also have a lot of respect for it.

This is just a thought I had while drafting my letter. Is it appropriate to find some sort of letterhead or something like that from the university (perhaps my major department?) to give more weight to my letter? 

I am personally an undergraduate student, but would it be appropriate for a graduate student or postdoc to use letterhead? Or should use of the letterhead be reserved solely for faculty and staff?
Do not use any professional letterhead when you are not speaking "on behalf" of the entity it comes from. 

Using your status, like explaining who you are or whatever, is fine. However, the sole fact of using entity's letter head, implicitly involves your employer and can only ends in a bad situation. 

If you are writing to another department of the same university, like in case of an internal matter, this can be deemed acceptable, but otherwise I would refrain from doing it.