Is there a keyboard chord for Context Menu button?

Shift+F10! One of my favorite shortcuts!

While not directly an answer to your question, there are possibly better solutions than Shift+F10. Just as an example, it doesn't work here in the Chrome browser, while I'm typing this answer. Also in most programs where I actually need the context menu key (aka Apps-Key) I use it so often, that a "use both hands or stretch one awkwardly" key comination like Shift+F10 is unfavorable.

Note however, that I can only comment on Windows solutions. Also these solutions assume that you have other keys, which you are willing to sacrifice.

1. Remapping Keys through the Windows registry (or SharpKeys utility)

Windows allows remapping keys through the registry. Sadly this doesn't use any straight-forward syntax, so you're probably better off using a third-party utility to generate the remappings such as SharpKeys.

The screenshot shows my configuration for the Thinkpad Edge E540, where the context menu key has been replaced by PrintScreen, but offers utility buttons (open calc.exe, lock screen, open browser, open "My Computer") which are entirely useless to me.

Thankfully it uses descriptive names for the keys, such that it is unnecessary to find out the keycodes yourself.

Remapping the keys directly with the registry would still require third party software in order to find out the keycodes (especially for non-standard keys), so I recommend not bothering to try it as long as SharpKeys works on your device.

2. AutoHotKey

Alternatively you can remap keys with more general-purpose utilities such as AutoHotKey. I used to use the script

; Print is Context, Shift-Print is Print.
$PrintScreen::Send {AppsKey}
$+PrintScreen::Send {PrintScreen}

to remap PrintScreen to Apps and Shift+PrintScreen to PrintScreen.

One weakness of AutoHotKey solutions however is that some programs, that access keyboard events on a lower level, such as many games, will not go through AutoHotKey.

Shift+F10 is such key. Unfortunately it's poorly supported in many applications (e.g. KDE4 Dolphin does not seem to support it).

As a workaround, I use a global binding that translates it into the standard menu key.

@kdb has explained how to do that on Windows, so you can use their solution there. On Linux I've got it to work with AutoKey.

Simply add this script and map it to <shift>+<f10>.

keyboard.send_keys('<menu>')

Note that you must save the script before the binding has effect.