Hi! I've worked on software that printed nametags for conferences as one-offs, and would dispute that perforated sheets "are just too cumbersome to work with in a production environment" if you haven't actually used them. If you're just
imagining that it would be cumbersome, I would recommend you try it out, not just once, but hundreds of times in a row, video taping yourself and timing each part of the process.
If you only printed on one card on a perforated sheet, it would still be worth the cost (250 cards in a package, 10 per sheet, 25 sheets per package, $12.50 package cost, so fifty cents per card to throw the other nine away) to use stock sheets and stock inkjet/laser printers. You can afford fifty cents as part of your per-user acquisition cost.
The suggestion by steanne to use plain card stock with a business card cutter is also a good one.
But, if you still want a dedicated device, you should decide if you want a paper-ish card, or a plastic one, like a credit card.
I would recommend getting a thermal receipt printer[1], which will let you print directly onto a small slip of paper, and then using a self-sealing laminating pouch[2] (not a thermal/heat-sealing laminating pouch). You print a slip of paper, slide it into the plastic pouch, it sticks together and there's your card.
For plastic cards, ID card printing machines are well-established for company ID cards[3].
[1]: Like this, but not DIY, they make normal USB and serial ones, too: http://www.instructables.com/id/Simple-and-Impressive-Busine...
[2]: Obviously a heat-sealing pouch would turn your thermal label black, rendering it useless: http://www.walmart.com/ip/Scotch-Self-Sealing-Laminating-Pou...
[3]: https://www.google.com/webhp?q=id+card+printer&tbm=shop#q=id...