Yes, you need to send a custom search to the API, passing in the email address as the correspondent axis.
So, once you've got a logon token:
https://example.fogbugz.com/api.asp?cmd=logon&[email protected]&password=SecretPwd
You can then do the search:
https://example.fogbugz.com/api.asp?cmd=search&q=correspondent:[email protected]&cols=ixBug,correspondent,sTicket,sTitle,dtOpened&token=cc83o7ri9c49t4vfvm3bn252ljvp23
Here I passed in a "q" parameter to the search command to set the search axis as "correspondent:[email protected]".
I also specified a number of columns I want to get back, such as the case number (ixBug), the correspondent's email address (I like to verify things like this), the ticket id the customer got as a response (sTicket, I wasn't sure whether this was what you wanted or the case number), the title (sTitle) and the date the case was opened (dtOpened).
Technically you don't need to ever ask for the ixBug column as it's always available as an attribute of each case element returned in the xml, but sometimes it's easier to have these things as an element.
There are lots of different search axis you can use, and many columns you can return.
Check out the search reference and the sample xml payloads at the botom of the API reference.