Hi Vicky,

As you have said, direct use of a browser from within The Grid has always been impractical - which is why many healthcare professionals providing EC/Computer Access have afforded browser accessibility to clients from The Grid via Rudolf Noe's MouseLess Browser (MLB) Add-on for Firefox.

The problem you pose highlights an issue the EC/Computer Access community faces with regard to its use of software which is not maintained.
The MLB Add-on for Firefox has undoubtedly benefited many clients, but at no time has Rudolf Noe been placed on a retainer, or similar, to ensure on-going support.

From the version history, I see that Rudolf Noe returned briefly to update his MLB Add-on in June 2016 before departing the Firefox community again.
It remains to be seen if Rudolf Noe returns once more to Firefox to update MLB ... but without any incentive to do so, then why should he bother ... ?

I did eMail Rudolf about 18 months ago but never got a reply.

If anyone reading this knows how to reach Rudolf Noe then please could you contact me with his details?

In the absence of an update to make the MLB Add-on for Firefox version 54 and later, suggestions how to proceed are as follows: 

(1) My initial thought is to play for time by installing the last version of Firefox that the MLB Add-on for Firefox worked with onto the client's computer - as per Will Wade's suggestion.

Judging from the comments, the Mouseless Browser Add-on stopped working somewhere between Firefox 50.1.0 - 53.0.3
I suggest you start with the penultimate Firefox version to 54 (53.0.3) and work backwards ...
All previous versions of Firefox are archived at:
https://ftp.mozilla.org/pub/firefox/releases/

NB: If you install an older version of Firefox then you will need to stop Firefox from updating itself by turning off auto-update (enabled by default).
To do this navigate to:
<Tools>/<Options[Drop-Down_List]>/<Advanced[Side_Menu]>/<Update[Tab]>
and then select the:
<Never_check_for_updates_(not_recommended:_security_risk)>
radio button.

Notes:
- Although by disabling updates there is a slight security risk - this can be mitigated providing Firefox is used in conjunction with (preferably) paid-for internet security & anti-virus software.
- As an additional precaution, it is strongly recommended that the Windows Administrator account not be used by the client, but instead a standard User Profile.
This means that in the highly unlikely event that the client's browser does get 'hacked' - this change makes it near impossible for a hacker to gain administrative privileges for a Windows computer and to alter important files.
NB: This only works if the Administrator account for the computer is password protected!
- Whatever precautions are taken, it should be noted that the security risk is likely to increase the longer this solution is in use.

(2) As the graphics cards installed in nearly all computers nowadays support 2 displays, the next suggestion is to use 2 monitors - one to display the Grid 3 user gridset and the other for the browser.

The additional screen space provided by using 2 displays (instead of one) should enable the use of letter combinations instead of numbers by allowing the entire alphabet to be presented on one display to the client.

Two VESA monitors having a 5:4 aspect ratio placed side-by-side are considered to provide the arrangement most suited to the user's field of view and which hopefully will require no, or minimal, head movement (which you have indicated is difficult for this user).

So long as both monitors are VESA then there should be no difficulty mounting them.

When using this method it is strongly recommended that the Windows desktop window containing The Grid 3 user gridset and the Windows desktop window containing the browser DO NOT OVERLAP - i.e. each window is confined to its respective monitor.

Also, as there may be a problem using a single eye-tracker across 2 displays, it is advised that calibration of the eye-tracker is limited to only the screen used to display the Grid 3 gridset.

Try "The Vimium Extension" ... which works in a similar manner to Rudolf Noe's Mouseless Browsing Firefox Add-on (but uses the alphabet instead of 0-9) described at:
https://www.techjunkie.com/browse-web-without-mouse/
... or else just type Vimium into a browser to go directly to an extension.

Vimium does work - however it seems that Vimium (when used in Firefox) may be less tolerant than the Mouseless Browsing Firefox Add-on, in that command sequences cannot be entered too quickly or too slowly or else they are not recognised - or at least this proved to be the case for the computer that I tried out this extension on.

A Vimium dialler gridset would need to be made for the The Grid 3.

Vimium is available for both Chrome and the latest version of Firefox.

(3) At present PCbyVoice are developing a software solution for Windows which may meet the need of this client - as it is intended to function in a very similar manner to Rudolf Noe's Mouseless Browsing Firefox Add-on - however the browser(s) that will be supported is/are yet to be confirmed.
When I last had a development update, around 2 weeks ago, the functionality of a prototype was being finalised.
Release of the solution proper is still a number of weeks away.

Hope the above helps - let me know how you get on,