is it possible to check if someone is holding down a key such as W or S?

Nope

It is.

A. by rebiding them to open a menu then make that menu move them forward (very laggy)
B. By holding a menu open the whole time that does movement for player
(also very laggy and players dont see the chat)

So, as it is possible, its not recommended.

You can check if player is moving though.

How would option A work? How do you rebind a key like this to a menu?

First exec rebind command through menu, rebinding it to either do 2 functions, or to do only one function (openscriptmenu), then add a waittill(menuresponse), whcich closes the menu, and does the action

Im lost, I did not understand well how is the way to do it.

"first exec rebind command through menu" where is to do that and how? Is there maybe an example at the game to take (quickmessages menu I guess) and which part of it?

I wouldn't do that, some people use esdf instead of wasd

What is the difference?

If the mod re-binded the players keys, since it can't detect what is being used for what. So players who use a different key combination from WASD would have issues as it would re-bind the movement command to the wrong key.

But this would affect esdf and wasd or not?