I got this solution to work in some Applications (like Finder, TextEdit, Firefox, ...) but not others (like Outlook, Chrome, ... )
A 'context click' can be mapped to a key/key-combo using all native Mac OS X features (thanks to its NeXT/OPENSTEP heritage). However, the solution is pretty technical "under the cover" stuff, and it may take some trial-and-error to get it to work.
Read about Key Bindings here, and follow (some) of the instructions to create your own ~/Library/KeyBindings/DefaultKeyBinding.dict file.
Key Bindings map keys (or key-combinations) to "Responders" inside the Text System.
I found the Responder for rightMouseDown here, and found the "magic string" for my desired target key (F13) from this url: osxnotes.net/keybindings.html
My ~/Library/KeyBindings/DefaultKeyBinding.dict looks like this
/* Context Menu */
"^\Uf710" = "rightMouseDown:"; /* Ctrl-F13 */
"^;" = "rightMouseDown:"; /* Ctrl-; */
Now I should be able to hit either Ctrl-F13 or Ctrl-; to get the context menu.
NOTE: You will have to quit & relaunch an Application to have it pick-up changes. (Worked for TextEdit in 10.10) Otherwise, log-out & log-back in.