Yes, it is most probably a problem that the Sony A7 is not listed in the officially supported camera list of Lightroom.
There are several things I found on the web that might help you.
Firstly make sure your camera's USB mode is not set to
Mass storage but to
Now you can try using other software to tether your camera.
Sony has the Remote Camera Control application available free of charge, and from the site it seems it supports the A7.
You can involve Lightroom in this process by monitoring the folders to which Sony's application imports photos. The photos will then show up in LR directly after you shot them so you can develop/inspect immediately. Here it's explained how to set this up.
If you really want to use Lightroom 5 to tether your camera I found a paid plug-in (web archive; original site is dead).
I am not related to the company, nor do I have experience with the plug-in, but there is a demo-version (web archive) available so you can try it out first.