When I first got this question from a friend I thought it was entirely the wrong sort of question. I wanted to write about noble, contentious issues of conservation concern, not bedbugs. I was wrong. Questions have a way of leading in interesting places.
My first step was towards Google. According to Bedbugger.com, (http://bedbugger.com/2007/03/19/faq-is-there-an-insect-that-will-eat-bed-bugs/) house centipedes, fire ants, and masked hunter bugs eat bedbugs, but none are useful as a treatment option as they are not available for purchase and could become pestiferous themselves. All three bite or sting. Now, I'd never heard of a masked hunter bug, and I don't like to take a random website's word for anything (nor should you, which is why I will always cite my source on anything I expect you to even consider believing), so I Googled "masked hunter bug" (yes, I like Google, and no, I don't work for them) and found a number of references, usually mentioning the hunters' fondness for bedbugs. The website of the PennState agricultural extension, http://ento.psu.edu/extension/factsheets/masked-hunter, gives their scientific name as Reduvius personatus, and an alternate common name as masked bedbug hunter. They are not native to the United States, having been accidentally introduced from Europe, and though they can live outdoors in the warmer parts of the country, they are largely creatures of structures, where they eat other insects who also like living inside structures. Incidentally, they are true bugs; while the word "bug" is generally used to mean any creepy-crawly, entomologists use the word to refer only to one order of insect, vaguely beetlish beings with sucking mouthparts and partially membranous wing covers--though the entomologist who told me that (see, I'm citing my source again), Jenna Spear-O'Mara, also uses the word "bug" like the rest of us do, on occasion.
Anyway, you do not want these things in your house; they bite, when bothered, very severely. Some of the websites called them "kissing bugs," which was interesting, as I'd read those can be dangerous. Another trip to Google revealed that they are indeed in the same family as the "kissing bugs" that can spread Chagas' disease in some countries, mostly in South America, but they are not the same insects (http://dermatology.cdlib.org/DOJvol7num1/centerfold/triatoma/vetter.html). Still.
But the claim that you can't use any of these animals to treat bedbug infestations bears a little more thought. Clearly, you wouldn't want to buy a jar of fire ants and dump them on your bed to get rid of bedbugs, but that sort of use of predatory insects (though helpful in gardens and quite preferable to chemicals) is a bit screwy anyway, if you think about it. I know it may seem so obvious as to not need saying, but insects and chemical pesticides aren't the same thing. If you think of your garden or your bed, or wherever else, as a thing that normally does not and should not have bugs, then bugs are an aberration, and you deal with them on a case-by-case basis by going in and fixing the problem. It's the same idea as taking an antibiotic when you're sick, or hiring a mechanic to clean the gunk off some unreachable part of the innards of your car. Antibiotics don't belong in your body--you don't get sick from antibiotic deficiency--and a mechanic doesn't belong inside your car, either, but you get your problem fixed and go on your merry way. Right?
Despite the common use of pesticides (and beneficial insects, which you can buy) in just this way, gardens, being alive, are more complex. Dr. Douglas Tallamy explained this very well in his very readable book, "Bringing Nature Home: How You can Sustain Wildlife With Native Plants." He describes finding a tomato hornworm--you know those giant green ones?--in his garden and deciding not to remove it because he could see it had already been parasitized by braconid wasps, who were eating the caterpillar's innards in the course of growing up. By keeping that one hornworm, he kept all the wasps who would grow up to kill more hornworms. But some of the wasps were being paracitized by another kind of wasp, killing them. So did he kill the other wasp? No; if the braconid wasps killed all the hornworms, there wouldn't be anything for the next generation of braconids to eat. If the garden was then recolonized by hornworms before the braconids got back, his tomatoes would be sunk. He needed the pteromalid wasps to keep the braconid wasps in check to make sure that there would always be hornworms to feed the braconids to ensure that there would always be braconids to keep the hornworms in check.
If I just lost you there, remember this; the goal in ecologically aware gardening is not to have no pests. It's to have a low, but fairly constant level of pests. If you ever get rid of all of your pests, you also get rid of all of your beneficials, and when the pests come back (and they always will) they'll swamp you. This is what I mean by insects being different from chemical pesticides; to use them well, you've got to have a completely different mindset.
Which begs the question; if we didn't have centipedes and fire ants and masked hunter bugs, would we have more bedbugs than we do already? It's a fair case?