Is it possible to fork a Gist to Repo via the GitHub site or client?
Not directly.
You can try to add a bookmarklet like "Fork your own Gist".
But there won't be any pull request feature.
If that becomes an issue, it is better to create a full-fledged GitHub repo out of your Gist collection of files.
(As in "Transfer gist repo into a github one").
Since my original answer in 2014, the ability to fork a gist (as mentioned in Bruno-Bronosky's answer) was added (around 2015).
Each gist indicates which forks have activity, making it easy to find interesting changes from coworkers or complete strangers.
![Gist forks -- https://help.github.com/assets/images/help/gist/gist_forks.png]()
See for instance "A basic collaboration workflow on Gist" (Apr. 2015) from Giovanni Cappellotto (potomak on GitHub).
Again, there is no direct pull request supported (still valid in 2020).