For a project I am working on, I need pictures of individual LEGO pieces, but with a transparent background. Does such a database exist?
I was unable to find any sites with a decent database of images in the file format you are looking for. I don't know what your experience is with image formats or what type of related editing software you may have access to that could help you obtain them yourself. If the list of parts isn't extensive then manually converting them is an option if you have the knowledge and software. There are a lot of sites like Rebrickable & Brickset that have an excellent database of parts with neutral backgrounds that would make a crop & conversion fairly easy. If there is a pre-existing database available that someone else answers with, I too will benefit from your question.
I did find this site that has online functionality and has the tool you need to convert pictures. I tried it out first to make sure it worked and it certainly does.
Here is an example of the trial I did with a picture from Brickset. Hovering over them should show you the different file extensions. (Win 8.1 w/Chrome it's on lower left corner of window) You can't really tell the difference unless you click on the pictures and/or save them.
As a last resort you could add them (part# & color) to your question. I would be willing to convert them for you, just not sure how to get them to you as this Q & A format would be inappropriate.