Snopes says incident is true because both of them have corroborated it. A simple google search leads to many links that were published recently. What I would like to see is original articles or comments that criticized her to understand 1) If it was prevalent in all of America or only certain parts of the country and 2) What was the tone and justification used for criticizing a simple act of inviting a black performer on her show?
I could have just asked a specific question as to what the original criticism was but I often feel like reading more from those times whenever I see a similar article, so I would appreciate if you guys could help in how a normal person would go about finding such things.
This was a matter of internal network communication, not a "letters to the editor" campaign or the like so you're very unlikely to find anything in a public newspaper unless you get lucky. The best way to start with this kind of research is to find interviews or memoirs with the people involved -- in this case, Betty White and Arthur Duncan.
Betty White has written multiple books, and fortunately, one of them (Here We Go Again: My Life In Television) not only mentions Arthur Duncan, but explains the whole story, and does a little debunking on the way.
From here, I clicked "Search Inside" and typed "Duncan" and was led to only two pages. Here is the full quote:
It came as a frightfully ugly surprise, one day, when a few of the stations that carried our show through the South notified us that they would, "with deep regret, find it most difficult to broadcast the program unless Mr. Arthur Duncan was removed from the cast." I was shocked, and it goes without saying that Arthur continued to perform on our show as often as possible. To its credit, the network backed us up. I was livid -- this was 1954, for heaven's sake! I wanted to tell the what to do with their stations, but wiser heads prevailed. To no one's surprise, that was the last we ever heard of the matter. They continued to carry us without another word on the subject.
You may note this does not match some other tellings of the story. For example, I've seen that NBC canceled the show specifically because of the incident, and you'll notice that the Betty White memoir indicates this was explicitly not the case, and that the objections were ignored.
I was unable to find a full memoir on Arthur Duncan's side, but I did find a newspaper story with block quotes. (Very important here: not a rephrasing, but actual words quoted said by the person.) I did not use any fancy research techniques, I just did a straight "Arthur Duncan interview" "Betty White" Google search and kept scrolling until I found direct quotes.
The time I was on (White’s) show, people from various areas resented black Americans on the program and they threatened to withdraw their support if I continued. I was not aware of these things taking place. I only found out about it several years later after Betty wrote her book. She explained it this way: ‘Needless to say, we used Arthur Duncan every opportunity we could.’ She stood up for her beliefs and that ended that. That solved everything.
Note he is referring to the book above, and he is paraphrasing Betty White's "it goes without saying". Again, though, there is no notion that the show was canceled specifically because he was on it, and Arthur even explicitly notes he didn't know about the complaints until reading about it.
This is not the only time this sort of incident happened -- I've written in the past regarding including Franklin in the Peanuts comic strip where Schulz specifically mentions a letter he received.
Interviewer: Do you think he was afraid that it would cause problems say in the South ...?
Schulz: Oh sure. There's no doubt about it. In fact, I did get one letter from one southern editor who said something about, "I don't mind you having a black character, but please don't show them in school together." Because I had shown Franklin sitting in front of Peppermint Patty.
I: Good God.
S: But I didn't even answer him.
In general I'd say it was good you were looking for the original story. Even when the story is "true" it can get a lot of distortion in re-tellings in an attempt to turn it into a just-so story with a clean dramatic arc. "There was an objection raised -> the network ignored it -> Betty White wanted to give the complainers a piece of her mind but she was talked out of raising a fuss" makes for a much less colorful story than "There was an objection raised -> Betty White doubled down -> it caused the show to be canceled".
There are tons of amazing archives for newspapers online. The key is to search for newspaper archives and once you find a good site start key word searching within there, rather than on Google. Here is an archive where you can set the date range and key word search for 'Betty White' or 'Arthur Duncan':
https://chroniclingamerica.loc.gov/
Beware you may end up spending hours reading old newspapers hahah they are super interesting.
I hope this helps!