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I want to draw a wavy border "rolling-hill" background. Specifically this is for flyers/business cards.

I think I see how I could do that by just drawing a line, and connecting the nodes, but is there any way I can do that by starting with a rectangle and skewing the top line of the rectangle?

First a caveat: For a biz card or a flyer, that is eventually going to a printer, a graphic created in a vector application such as Inkscape is preferable.

However, this is a Gimp forum, so a Gimp reply.  Use the paths tool to create the curve, then path to selection and fill the selection.

Three and a half minute example: https://youtu.be/bxwIIQeDgqI