The Balanced Corpus of Contemporary Japanese (BCCWJ, http://www.kotonoha.gr.jp/shonagon) draws on published sources such as literature, newspapers, etc., as recent as 2005. The numbers for the words in question are
移民者{いみんしゃ} iminsha 8 results
移住者{いじゅうしゃ} ijūsha 181 results
労働者{ろうどうしゃ} rōdōsha 6982 results
定住者{ていじゅうしゃ} teijūsha 19 results
在留者{ざいりゅうしゃ} zairyūsha 25 results
永住者{えいじゅうしゃ} eijūsha 91 results
帰化者{きかしゃ} kikasha 1 result
The only result for 移民者 in online J-J dictionaries (accessible via http://www.kotobank.jp) is the title for the Swedish film "The emigrants" 移民者たち.
That said, 移住者, 定住者, 在留者 and 帰化者 are not in any of the dictionaries either.
I'm not sure what conclusions to draw from this. 移民者 is certainly very, very uncommon, but not unthinkable. That said, 帰化者 seems to be even less frequently used.
I see several problems with the argument here.
One problem might be that 移民 literally means the moving/migrating of a person, which makes 移民者 look like a little pleonasm, so that 移民者 might not exist because it sounds silly, not because the Japanese don't accept the concept of immigration.
Another problem is that all the other nouns (without 者) can used to form a verb with する, e.g. 移住する, 永住する, etc., which lends itself very well to form, say, 永住者 meaning "somebody who does 永住". Only 移民 doesn't have the corresponding 移民する.
Another thing I have to take issue with is describing the word/suffix 者 as "honorific". I don't think anything justifies this description, especially since "honorific" means something very specific when talking about Japanese, and that is suffices like さん, 様, 殿, etc., usually used for addressing someone (maybe the equivalent being Mr., Ms., etc.), not for describing someone.