From this question I learned this syntax:
https://stackedit.io/viewer#!url=http://path/to/markdown.md
However I did not find how to open a local file (which is possible with the "Import from disk" dialog).
Is is possible to open a local document with a similar syntax to:
https://stackedit.io/viewer#!url=file:///C:/test.md
Unfortunately, the app does not handle any (download) protocol but http and https.
If you just wanted to access them (not edit and save) you could run a simple static file server you could access them, just not save them. Likely useless, I know, but here for completeness.
Because the app is hosted in a browser, you wont have real access to your local File System (except through Dropbox/Google Docs, which use an API). You can see that the app also has a local version of an MD file included, but again, read only.
I am sure there will be someone who might host it in electron, which would give you complete File System access with a few minimal tweaks.
I for one would love to integrate this into an internal documentation server. Along with my thousand other projects I want to work on...