You might get some snow but it's unlikely to accumulate. My birthday is just before Halloween and historically the first snow in NE Ohio generally hits between my birthday and the second weekend in November, but it's almost always a big lot of nothing that barely sticks to the grass let alone the roads. The last decade it's only snowed twice in that window, I've gone camping my birthday weekend more years than not since we got our first trailer 6 years ago.
What you might have some trouble with is finding campgrounds to stay at, many places in the North close at the end of October. Your best bet is state campgrounds where you probably won't have water but you'll have electric and access to a dump station.