As Quintessence has already said, the fanmail address for writing is:
c/o Wind-up Records
72 Madison Ave
New York NY 10016
If you want to write, you must put United States at the end of the address if you are sending from outside the US. I would also strongly advise adding your address as a return address in case it is returned to you as unsent or unable to be delivered. If you want to be sure it's got there, pay for a registered (signed for) or a fully tracked mail service, then you will be able to tell if it has been signed for/delivered.
The email address given is:
I can't confirm if you'll get a reply, but as a general rule fanmail sent by letter is often forwarded onto the band or their manager. You're more likely to get it read or possibly get a response from someone if you write. If you email you may get a generic response back.
Original source of this information:
I would strongly advise against trying to contact Amy Lee through her personal details [if you should find them] or by contacting Amy Lee's friends or family [whose details may be easier to find] as it can be quite intimidating to have a stranger phoning up you or your family. Not to mention that in a worst case scenario if you keep doing it, you could well find yourself being spoken to by the police or getting a letter from a lawyer over stalking/intimidation.
Right now, I would say your best bet is to try and meet Amy Lee and the rest of the band whilst they're on tour as they're doing meet and greets and instore signings.
You may find getting access to these sort of VIP events easier if you join EvClub as they often get informed and get given the chance to get tickets to meet and greets before anyone else.