The only thing I can think of, if your full name and address aren't relevant to whether you're eligible, would be to fill out the form with something very obvious like "Jane Alias, 1600 Pennsylvania Avenue, Washington, DC" and include your explanation in any free-text field for explanations/more information. That might work if eligibility is based on something like the person's income or age.
However, a lot of organizations that offer discounted services have eligibility based on where people live. For example, I went to a hospital emergency room a few years ago, and the paperwork we were given included information on applying for charity care if I lived in one of a few specific zip codes near the hospital. For something else, it might be a broader "resident of King County" or "do you live in New Jersey?" So while you could put a friend's address (with permission), it's possible that your actual address is eligible and theirs isn't: or vice versa, in which case you might get the appointment, fill out the form more accurately, and be told that you weren't eligible.
What you want to avoid is making the person at the other end think "I suppose lilblu might be eligible, but they're going to be a pain to work with, I'll wait and see if they fill out the proper form." So the closer you can come to doing things via their standard process, the better your chances are; remember that you're probably making extra work for the person at the other end. If their system can't make/record an appointment without a properly-formed name and address, if you don't give them something, even "Jane Alias," the employee has to decide whether to go to the extra trouble of filling in the form for you, with a fake surname and address.
Also, with regard to legitimate reasons for needing your phone number: the telephone is a good way to arrange appointments if you care at all about when they see you--and almost everyone does, either something is too early or too late, or you have two other appointments between now and the end of June but this conflicts with one of them.