Short answer: Yes, actually. I think. Maybe. It probably depends on the person.
I have no scientific studies to back me on this, but it's entirely possible that that's because there... are no scientific studies on this. I wasn't able to find any, at least. (If anyone actually knows of any, please post them!) So I'm going to Albert Einstein this with a Gedankenexperiment, with a fewadditional websites thrown in that talk about gastrointestinal issues and gas and bloating, so take what you will from it.
Whenever one chews or drinks, one is inhaling air into the stomach. That's just the wayingesting food works. That said, people really don't inhale too much air in this fashion, but that ingested air is the primary reason we burp and, to a lesser extent, flatulate (possibly the best word ever). The amount of air we ingest can be increased in a variety of ways, basically any time one actively takes more in:
Gulping/sipping/slurping (sipping hot coffee, one can hear the air being ingested)
I'd be inclined to say, however, that straws are probably not the biggest culprit on the list. Whenever one drinks from a straw, it seems like there's really only a minimal amount of extra air being swallowed (whatever is present in the tube above the liquid line). (Extra air on top of that isn't being swallowed, as the mechanism for drinking through a straw is actually due to the drinker relaxing the jaw, which increases the amount of volume in the mouth/throat, which decreases the interior air pressure. Outside atmospheric pressure then forces the liquid up and in.) A confounding factor for your friend might be that a common beverage taken in through a straw is soda, which will definitely add more air (carbon dioxide) to the stomach, possibly causing feelings of bloating. Or if she's convinced of this, it might even be a placebo effect at this point.
So to sum up: yes, drinking through a straw causes "excess" air in the stomach. But probably not too much, as regular eating and drinking will do that as well, as will many other daily activities (as I noted, slurping hot coffee is probably a much bigger/more frequent culprit than anything else). So tell her she's right that it'll put air in, but it's most likely not that big of a deal. Live a little! Use a bendy straw! Burp!
(I apologize for being unable to back any of this up with hard numbers.)