If a girl has amenorrhea (doesn't menstruate) can she become pregnent?

That depends on why she isn't menstruating.

Tubal pregnancies can occur, in which the egg is fertilized in the Fallopian tubes (can be very dangerous for the girl).

But if you are asking if you don't need to worry about bc because a girl is not menstruating then I should tell you YES, you should always worry.

What if menses can occur, but only when on the pill? Possibly indicating some sort of hormonal imbalance?

Then a visit to a gynecologist is in order.

Absolutely, see a qualified gynecologist, preferably at a good hospital.

Can you rattle of the names of a few good hospitals. I've been witness to many 18th century Drs where I live.