Whoops! Sometimes when I open my mouth it is only to see how much FARTHER I can stuff my foot into it. Calling native Americans "Newcomers" rates at least "up to the knee" on the "how far?" scale.
The LDS 1880 is a transcription of the 1880 US census, not a book. It is available on-line courtesy of the LDS Church. It took them 17 years and a couple of million people-hours to get it transcribed.
I just tried it again, with Given name blank Surname "Kaye" exact spelling box not checked, Race = "Native American".
I got 74 people in the US, none in the four corners area.
Your family may not have been counted, or they may be under a name that the search engine on the LDS site doesn't equate with "Kaye". I tried it with the most common name I could think of as a test:
Given name = John Surname = blank State = Arizona Race = Native American
I got 56 hits, which is small. I'm not enough of an expert to know if they counted people on the reservations or not. I know Oklahoma isn't there.
Going to newer people,
http://ssdi.genealogy.rootsweb.com/cgi-bin/ssdi.cgihttp://ssdi.genealogy.rootsweb.com/cgi-bin/ssdi.cgi has 34 people who died somewhere between 1960 or so and 2004 in AZ with the last name Kaye. Click on "Advanced Search" to set the LR or LB to the state you want.