Whatever you're after - from standard bulldog badge clips through to an easy Breakaway safety lanyard - you'll have no problems finding a manufacturer that can do this. What's more they can do this at a very fair price.
Most of the online suppliers who specialize in printing up id badges or supplying id holder wholesale orders will offer a decent range of attachment methods. The advantage of an alligator clip is that it does the job reliably and without fuss.
Of course if you want something a little more exciting (for whatever reason), there is a world of choice at your fingertips. In the arena of creative name tags you have the beaded chain id badge that comes with beaded lanyard patterns right through to the smart and efficient (and very long lasting) black neck lanyard style. While you're sourcing an appropriate company you should take a look at the other similar products on offer that may still meet your price requirements.