I think my art would look appealing on a holiday card. Do any nonprofits accept art submissions?

Several do, though some ask that you live nearby or have a connection to the charity.  Charities that accept art submissions include:

Courage Center – desired themes are Thanksgiving, winter city scenes, winter landscape/wildlife, Christmas traditions, and religious Christmas/peace; deadline is November 30th of the previous year.

Dana-Farber Cancer Institute -“artwork or photography featuring Boston Red Sox or New England Patriots-inspired images, winter scenes of familiar Boston landmarks, or New England beach scenes and lighthouses in .jpg format”; deadline is mid-May.

CASA of New Hampshire – (must be a New Hampshire resident); “seasonal-themed drawings, photographs, or paintings”; deadline is July.

Hunger Task Force of Milwaukee – (must be a Wisconsin resident); deadline is June.

Some children’s hospitals, such as OHSU Doernbecher Children’s Hospital in Portland, Oregon, accept submissions from children who are or were patients.