I was looking at using GravCMS to build a site on my GoDaddy shared hosting account.
Is there a way to do this without having root access to the server?
That looks to be a fairly straight forward application and looking at the requirements I don't see anything that would require root access.
I would suggest using our cPanel web hosting for it.
If you are having trouble installing it, feel free to follow up with any errors you are seeing, and perhaps a member of the community that is familiar with it can offer some suggestions.