Retrieving contacts from Android phone (Sony Xperia Z)
I had a similar problem on my Android Tablet, not sure if my solution will work for you.

I had a Chinese Android Tablet where you can see the screen but the touch was not working. I used a USB mouse by connecting it to my tablet using a cheap USB Female - Micro USB Male connector. That allowed me to use the mouse as an alternate input.

If this is possible on you phone, you can export your Contacts to a SIM or SD card.