Firefox:
What you want seems to be possible according to this answer, but I haven't investigated it closely.
Chrome:
Outside of an NPAPI plugin, this is definitely not possible. Your browser cannot access the filesystem except when the user explicitly asks it to do so via a file dialog. If any page on the Web were able to programmatically set a filepath and then fetch the file when the user clicks a form button, that would raise massive security concerns.
So, a normal webpage certainly can't do it, obviously. If a Chrome extension could do it (i.e. read any file in the filesystem) then it would at the bare minimum require a special permission. The list of permissions does not include filesystem access, so that's out of the question.
The only option, as I mentioned in the beginning, would be to develop an NPAPI plugin to the read the file for you, which would be like building a hospital to treat a pimple. NPAPI has a pretty steep development learning curve, and it introduces significant security concerns (since your extension can now act the local user, the cost of a security compromise is much higher).
If you're willing to place the file on the browser's virtual HTML5 filesystem, then you could do that. Without specific knowledge of your needs, I can't say if that will be appropriate for you.