The distinctive, comical sound of the kazoo is one that can't be fully imitated without the instrument itself, but when you're in a pinch, you can do a vocal reproduction that will successfully bring the sound of the kazoo to mind. This technique is great for show choir performances or just the occasional round of sharing novelty human talents.
Purse your lips. Form your lips as if you are about to make a "V" sound with your upper teeth resting on your lower lip. Make a sustained fricative "V" sound (meaning you leave off the the "ee" you would speak if you were pronouncing the letter name). This is the start of the kazoo sound.
Place two fingers across the opening of your lips. Press the palm side of your first two fingers at a diagonal across your lips, as if you were taking a cigarette or toothpick from your mouth. Keep the fingers pressed gently together; you want them to vibrate slightly while you make the kazoo vocal noise, imitating the sound of vibrating wax paper.
Shape your mouth to produce a kazoo-like vocal sound. You want to make a very nasal sound, so shape your mouth as if you were forming a letter "e" (tongue raised in mouth). Form the back of your throat and oral cavity into a tight, high opening.
Imitate kazoo breath attacks. With a real kazoo, the start of each sound is very slightly delayed as the air pressure builds up, then starts to vibrate the wax paper. To imitate this without a kazoo, start every sound with a breathy "h" sound. Let it hiss for a tiny moment before you start the vocal sound.