I want to be able to create an .ico file containing a complete set of Windows icons (16x16, 32x32, 48x48, and 256x256). Is there a web application that can do this for me given a png or bmp that is already 256 by 256?
You could try with ConvertICO.
You can select as many output sizes as you need.
Another option is IcoConverter.
The aspect ratio is not preserved, so squarish images are recommended.
If you do it yourself use gimp.org. like Adobe Photoshop, but free. Photoshop does not come with a way to combine a .ICO file or to create it. You can purchase a plugin, but it is risky and Photoshop CC wont work with them.
Gimp allows you to combine and save .ICO files. And it is a free piece of software.