The Grand Canyon is around 6,500' elevation at the south rim, and 7,500 at the North Rim. So they close the North rim around October, depending on the first big storm, they do not plan on plowing the roads out of there (67 miles to Jacob Lake). They get feet of snow at the North Rim, and normally the roads are not open until the end of April, but it changes year to year..
I found out all about driving on ice back in 1987 when I thought "Thanksgiving is a great 4 day weekend to try out my new to me camper - lets go to the Grand Canyon!" And I got to see a lot of snow and ice. The canyon was foggy, but the road was open to those who wanted to get there. I had a friend check their newspaper for weather temps at Flagstaff AZ, looking for a good time to go back, it did not get above 60 until late spring (7,500' elevation).
My uncle was telling me about a school bus driver from Pennsylvania who was stuck in the snow around Flagstaff in June, and was the only one with chains in his car, so the highway patrol said he could drive in the snow. So to answer your question - some Octobers can have snow, and in May too. But just a few miles south in Prescott, or Congress, at a much lower elevation it can be fine weather.
Check out the Congress Escapee's RV park. Very large lots, and you can rent them for a low cost per day or month. There are Escapee's RV parks all over the south, near Yosemite, and one in Oregon and one in Washington. They are basically built for full timers by full timers. THey have a mail forwarding service, and other things you might find useful while full timing.
Fred.