In the notes I am using, it states that if two observables A and B are measured simultaneously, then the measurement of A does not affect the measurement of B, and vice versa. However, why does the frame of reference seem to be neglected in this explanation? For another observer, it seems that if A and B are measured simultaneously in the first frame, they will not be simultaneously measured in the second frame; thus the measurement of B should affect the measurement of A? Is this due to some change in the observable due to the relative motion that I am not aware of?
In relativity there exists a notion of a "space-like separation", that is, a separation between two events that cannot be viewed in any frame as happening at the same place, only at a different time. Additionally, you can show that events at these separations cannot be connected by light signals. Last but not least, there will be a frame in which these events happened at exactly the same instant.
That is, if we observe an event $A$ influencing an event $B$ at a space-like separation, there will be a frame in which this will look like instantaneous action at a distance, and in every physical frame this will look as an influence propagating at superluminal speeds. In other words, for causality, you have to require no influence of measurements over space-like separations, and this is indeed one of the building blocks of quantum field theory, the relativistic extension of quantum mechanics.