Sorry, you have to have administrator privileges to install Paint.NET.
You can, however, install it on one computer (say, your home computer) and then copy the contents of the installation directory to another computer (your work computer). It will work, assuming you have .NET installed, but you will lose functionality. You will not be able to print, scan, or acquire from a camera, nor will the shell integration be present (file type associations, and thumbnails for .PDN files).