Is it possible to add an internal broadband card to MacBook Air? I have a Mid-2011 MacBook Air 11". Can I add a Verizon (or any kind of) broadband card to it internally? We want to avoid having an awkward USB modem hanging off the side.
No unfortunately the only options are really external. Until someone can come up with a 3G / 4G card that will work in place of the MacBook Airs Wifi / Bluetooth card, but even then how portable are the antennas designs between Wifi and 3G / 4G cellular signals, not sure it would work well if at all.

See iFixits teardown of the MacBook Air 11":
 

A mobile WiFi Hot Spot is a real option for now, essentially a portable broadband card that interfaces with your Mac via Wifi to allow a data connection. Also if have a smartphone such as an iPhone or Android phone that offers this Wifi sharing you can use one of those as well to do the same thing.

These kinds of devices have their own battery and would allow  you to tuck them away in a backpack, bag, pocket etc and use your MacBook Air with out any dongles, etc.