If you can find one with two "keyhole" style holes in the back you can mount it vertically (portrait mode) by placing the mounting screws into the wall in the correct orientation and sliding the frame onto the screw heads from the side rather than from above. It will be a little less secure than in the intended landscape mode, but a small piece of double-sided foam tape placed between the frame and the wall near the "bottom" edge of the frame (when in landscape mode) should be enough to hold it in place. Just be sure to keep that edge of the frame tilted slightly away from the wall until the "keyhole" is slid all the way onto the heads of the screws in the wall.

You may also have to manually rotate all of the images on a computer prior to loading them onto the frame, or onto a memory card to be used in the frame. I doubt a frame that is not made specifically to be displayed either way will be able to do it automatically by sensing which way is down.

Having said all of that, it took me all of about 15 seconds to find these Nix photo frames with mounting keyholes for either orientation, offered in four sizes from 8" to 15", on amazon.

B&H offers a few similar frames as well, including this 12" digital picture frame.

No longer than it took me to find these, I would think that they are not that uncommon at all. Be sure to include the word "portrait" in your search.