GoDaddy might not know it to ask them, but they do support ZenCart, as a matter of fact, in their "GoDaddy Hosting Connection" (accessible if you have GoDaddy Hosting) Zen is one of only 2 eCommerce solutions they deploy via script.
I was talking to tech at GD yesterday and they are scripted to tell you that they don't offer Technical Support for Zen, but the GoDaddy site DOES support Zen. (v.1.3.8)
I'm running it there right now with no problems, however, it is my own install... (The GD scripted install never really seemed to install correctly.)
What I won't recommend is doing what I am doing. I developed a Zen Cart site on my hosting account with 1and1 and have now moved it to the clients hosting space on GoDaddy. YIKES! It's hard to tell which of them has the most proprietary system! (You won't find a standard cPanel in either.)
So, the two lessons learned here are...
- Zen is going to work and perform fine on just about anyplace you can install it.
and...
- Don't EVER AGAIN consider devving on one host and moving to another later.