In Riemann's definition of integration in $[a,b]$, a step in the process consists of choosing one point from each part of the "current" partition for further a partition, and again choose one point from each part where the function will be evaluated.
Is this mathematically\logically possible without the Axiom of Choice?
The axiom of choice is not needed at all. Note that the integral is defined as a limit over finite partitions. Moreover these partitions are partitions into intervals, so the partitions themselves are simplistic as well.
The partition are finite, so there is no need to worry about the existence of choice functions, and we don't choose a particular sequence of partitions. We consider all of them.
And as the old saying goes, if you don't know what to choose - take everything.