R, 79 bytes
f=function(x){n=length(x);for(i in 1:n){j=sample(i:n,1);x[c(i,j)]=x[c(j,i)]};x}
f=function(x){n=length(x);for(i in 1:n){j=sample(i:n,1);x[c(i,j)]=x[c(j,i)]};x}
This is a straightforward implementation of the Fisher-Yates shuffle. The R function sample draws a simple random sample of a given size from a given vector with equal probability. Here we're drawing a random sample of size 1 at each iteration from the integers i, ..., n. As stated in the question, this can be assumed to be O(1), so in all this implementation should be O(N).