My duaghter hit a snow bank and damaged the radiator and the mechaninc told her that the frame was bent too.?

Its so hard to say without actually seeing the car. A radiator is something or nothing to be replaced but it depends which part of the car chassis he means. If it is bent or twisted in anyway it should not really be driven but people don't always abide to that i know i don't. He may mean the slam panel (a big metal girder running behind the bumper) or he could mean subframe damage either way i personally would get it checked out.

What does alignment have to do with this question?
It depends on which frame is bent, if the radiator can be supported and you should no have a problem. Depending on the car, radiators sit on some kind of a frame, I am guessing that is what he is talking about, but in most cases it can be fixed pretty easily, unless its broken. If its the body frame, it is not something that would bend easily I dont know if a snow bank would bend the frame. If posible get a second opinion especially if he is asking for alot of money.
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No, not safely. Sure you could replace the radiator and drive it, but with the frame bent, it will drive really strange and be very dangerous. I wouldn't let my daughter drive it.
At the least, tires and other parts will wear out very quickly as they wont be aligned anymore, and the car will drive erratically.
At the worst, The damage may cause stress on vital parts like brake and fuel lines, causing them to fail suddenly, causing a flaming wreck. Flaming wrecks are cool in the movies. In real life, with your daughter driving, not so much.

I would be willing to bet that he is talking about the radiator support which is a welded on component that supports the radiator etc. Depending on how bad it is, you might be able to just replace the radiator. If the radiator support is severely damaged, then it must be replaced in order to replace the radiator.

if the frame is bent up kissing the engine then the radiator won't fit 'cause the radiator goes in the front. if it ain't bent so far then your good but don't speed with a bent frame.

i doubt it that the frame is bent. it might be out of alignment. change the radiator and take it to alignment center and they can tell if the frame is bent. he was probably talking about the radiator support was bent but you can straighten it out.

You would most likely go through tires alot faster for the fact that it would be out of alignment

Your daughter just cost you money because she sucks at driving. Teach her how to drive.
Should work if the damage is not that bad.

possibly but you would have nothing but problems