Those are legitimate questions.  As others have posted,  the first step that will happen is the US Treasury will deposit this $1200 by direct deposit, for those whom are set up to receive direct deposit.  Not everyone is set up with direct deposit with the US Treasury.  So, the next step will be to mail out actual checks for those who do not receive direct deposits.

Regarding the amount of cash kept in a bank, you are correct - banks only have a certain amount of actual cash on hand.  If everyone went to the bank and demanded their money in cash, that would be considered a "run on the bank".  That is what was portrayed in the movie "It's a Wonderful Life".   That has happened in the past.  If there is a run on the bank, the bank would close down, the Feds would step in and provide financial support to get them thru the crises. 

To answer your question, I believe I did hear that the Fed will print money - actual paper money - as needed to get this $1200 into the taxpayers hands.  Since the dollar is no longer based on the Gold Standard (and hasn't been for decades), and is based on the Faith and Trust in the US Government, they can print as they see fit.  However, as mentioned, much of it will be electronically transferred.