You could email your desired itinerary to
[email protected] and they will handle the booking, pricing via the AA rate desk, ticketing on AA ticket numbers, etc. In order to issue the ticket, you will need to email your CC and passport details -- they will provide a list of the required details.
The only caveat is that you will need to pay a service fee of EGP 200 in cash or wire transfer.
I used Emeco 3-4 times in the past, even visiting their office (Taquir Square or something) once. They were lovely. With the old paper tickets, they would even hold my paper ticket in their safe & deliver the ticket to my Cairo hotel when I arrived to start my RTW.