How do I rearrange my list of banking accounts on the far left side of the register window?

To sort or alphabetize the account bar you have to open the account list (go to Tools - Account List or Ctrl+A) and highlight one and move it up and down one at a time.

 

In a new Quicken file, new accounts are listed in alphabetical order (within their account group) ... until the user manually re-orders any of them. After that, there is no way to get them back in pure alphabetical order except by manual re-ordering, one account at a time.