First, call back and try again because technically these numbers for businesses are public information, so there isn't a legal reason they can't give you the Federal Tax ID number over the phone (it might be a company policy, but maybe you can get a supervisor who is willing to make an exception). They should also have a mechanism to provide you with a replacement or give you contact information to their tax processor who produced the form for a replacement.
Otherwise, your best resource is going to be an internet search. I can't put links to casino sites in this answer but I was able to quickly use Google to find all of Harrah's casino ID numbers across the country directly on their websites. If its a smaller, local casino, this information may not be on their website, but sometimes even the small casinos are owed by these larger ones.
Next, there are online services that will do public records searches for Tax ID numbers (these numbers will be listed on public company filings), but they will likely cost money. You could try to use the Securities and Exchange Commissions database known as EDGAR to do your own search for the company filings. However, this SEC database is not very user friendly, but I have included a link below if you want to try.
https://www.sec.gov/edgar.shtml
Lastly, if you know all the other information, you might need to just file by paper mailing. You can try to e-file without the ID number but you will likely receive an error or rejection. And you don't want to file without the W2-G information because that will generate an IRS notice.
