In crude oil gray market purchasing, proof of product is often
verified on the Shell Screen in London.  What exactly is the Shell
Screen in London, who operates it, and how do you get access to it?

Two dollars doesn't buy much in the crude market  - and not much here,
so I will try to help with free comment that may lead to a full
answer.

From the following excerpt from a site that gave a description of
buying Nigerian crude oil, I ASSUME that "shell screen"  and  "NNCP
screen" are websites of  Shell Oil (which is active in Nigeria) and
NNCP,  Nigerian National
Petroleum Corp., on which shipments of crude oil from Nigeria are posted.

"6. Lifting: Exactly (?) seven days after the confirmation of the
buyers LC, seller shall furnish buyer with reassignment of the cargo
in the buyers name including the loading date and the tanker loading
approval which shall be confirmed on tabloid screen via: Shell screen,
NNPC screen. Also [CIF] Seller shall engage the services of the
vessel: [ vessel name, type & registry] for the delivery of the cargo
at buyer?s destination port, seller shall also furnish buyer with the
charter party/vessel particulars."

http://66.223.105.237/ubb/Forum1/HTML/000546.html

Here is the Shell "directory", but  - if my assumption is correct - 
it doesn't seem that the "shell screen" can be accessed from it:

http://www.shell.com/home/Framework?siteId=media-en&FC2=/media-en/html/iwgen/zzz_lhn.html&FC3=/global/shellworldwide_shared/shellwebsites.html

Good luck, Myoarin