Ok - First take a look at this information:
How to get company logo from Linkedin API?
Then take a look at this web page:
https://www.linkedin.com/nhome/updates?topic=5967002483262787584
The other problem is similar to your problem. You will have to scrape the web page to get the information. If you look at the <TITLE>...</TITILE> part on the IBM web page you will see it says "IBM | LinkedIn".
That is how you get the company's name off of the web page's HTML.
The LinkedIn API says:
https://developer.linkedin.com/docs/fields/company-profile
Check out their Profile Information. Note the "name" field? The company's name should be in that field also.
(BTW: I do NOT work for Stack Overflow or LinkedIn. Just in case you were wondering. I'm just a good samaritan helping out. :-) )
Update: I just re-read your request. If you know what the company's name is you should be able to use the API to reload their profile which has their ID number on it. Don't know about an update key. Do you get what I am talking about?