Why can you not use fans? No electricity? But how about light? Is it the sound? If you run a 12V computer fan on 7 or even 5 V it is extremely quit! It will run on lower RPM and will be less effective in moving air but it will help a lot.
Do not buy a small fan! They are usually more noisy and does not work well on low voltage (move to little air), use a 3 or 4 inch!
One other "bad" option is to create a draft through the cabinet. have ventilation on the top AND in the lower front, see
The Orchid Source Forum: Constructing a Orchidarium - Step by Step (warning: multiple Illustrations). The hot air created by illumination will raise and new DRY and cooler air will come in through the front. IF you manage to balance everything it will work when the light is on but you will get stagnant condition during night!
You may get BIG humidity problems this way......