My experience of the main "central" Garanti:
You don't need a card to get a ticket, use the machine just inside the door on the left hand side next to the security guard. There are 3 options for different tickets, I always took a ticket from all 3 as could never work out what did what!
The free phone on the right hand side will connect you to English speaking customer services, I think you dial 9 but the instructions are there for you.
Failing that contact Alper (username Banker on this forum) who now works at Garanti & I'm sure he will help.
Edit: The lady who sits at the far end in the customer services section used to speak perfect English too if all else fails go straight to her maybe?