Your solution is to probably buy at an office supply store (using an Ink credit card) sheets of business cards (there's an Avery product and compatible store or generic brands) that will feed through the printer (as regular 8.5x11 sheets, and then you can easily separate the cards (8 per sheet IINM) from the printed sheets.
I would suggest that you don't waste your time trying to find a desktop printer that can feed pre-cut business cards through. It's very difficult from a technical perspective, and IF you find anything I would guess it will be costly.