With the changes in the economy, I'm inundated with resumes. About a third use the shotgun approach: apply for anything and everything.
I recommend being more strategic. Honestly, unqualified resumes clutter up my in-box and do nothing but cause me more work.
In addition, my standard "How to Apply" line is: "Send cover letter and resume to kfrancis@. I've noticed that many applicants who don't meet our qualifications also neglect to send a cover.
You're shooting yourself in the foot when you skip the cover when asked. One: you have shown that you don't follow instructions--even when the stakes are high. Two (and even more importantly): you have missed a golden opportunity to market yourself.
If you don't have the experience or the education, but you're perfect for my job, how will I know it unless you explain? Most nonprofits are mission-based; explain how you are passionate about our mission, and at least you have a chance. Skip the cover, and I don't mean this unkindly, but the opportunity will be skipping you.
So see the cover letter as an opportunity rather than a chore...and best wishes with your job search!