Getting custom printing of arbitrary shapes on arbitrary materials is hard to do, even in this internet age of CafePress and print-on-demand book publishing.
One thing that's easy, though, is making custom stickers: go and Google "sticker maker", and it will show you sticker-making machines. Those machines take any shape of cut-out paper, and make the back sticky. Print out your own custom character card with your favorite image-editing software (possibly scanning in the Betrayal at House on the Hill character boards first for a template?), cut it into a pentagon, cut out a pentagon of tagboard, then use the sticker-maker to make your print sticky and stick it to the tagboard. (For a better fit, make the sticker bigger than it needs to be, and don't cut the tagboard until you've put the sticker on it.)
This is morally equivalent to getting a Glue Stick and making the sticker yourself, but I've found that the glue in sticker makers is generally better than glue sticks, and sticker makers will apply the glue precisely to the edge of the paper. Depending on exactly what thickness of tagboard you find in your local art supply shop, you may need to layer them to get the right thickness for the clips that come with Betrayal at House on the Hill.