Ok, maybe the first response was curt. I just don’t find excuses as you offered acceptable with all the tools available.
So to be constructive, may I offer this possible solution to your aspiration?
First, you should put down on paper any ideas, goals, services, activities, etc., the overall vision you would like to see become of this club. Brainstorm with a couple of your friends that are already interested in photography. Approach the president of another club in your school and query him/her of the processes they go through to keep the club running and supported by the administration. This will give you some idea of the hurdles you will likely face.
Next, I would take a poll of your school mates and get a general idea of how attractive your idea of a club would be to begin with before you approach the principal. He’s an administrator and numbers are likely to be an important factor for his decision making. With this poll, make yourself a list of general questions that would not take more than a minute or two to ask and respond. You’re attempting to get just a general idea. You can make it as simple as going up to a student and ask “Hey, Joe. I’m trying to find out if we can get a photog club started here at school. Do you think that would be something you’d be interested in?”. If he says No, say thanks and move on. If he says Yes, whip out your clipboard and jot down his name. Then I’d ask a few questions such as 1) would you be an active participant? 2) do you have a camera? 3) how much experience do you have? 4) what type of shooting do you like to do? If you make it in a grid table form where you can quickly check a box or write in a code, so much the better. You don’t want to drag this out into a discussion or debate.
With the raw data you’ve collected, compress it down to usable information that you can put in your letter to the principal. I would also try to find a student that excels in English and composition to assist in drafting the letter. If you do the leg work, propose a reasonable intent and provide benefits that this club will bring to the school as a whole, I think you have a better than average chance to pull this off.
That's my nickel's worth. Good luck.