JesseM writes: No, even if A takes into account the light-speed delay she still won't conclude that B was moving faster than light. For example, A might see an image of B that appears to be 4.999 light-seconds away at t=9.999 seconds, and conclude that since the light took 4.999 seconds to reach her, B was actually 4.999 light-seconds away at t=9.999-4.999=5 seconds in her frame. Then A might later see an image of B that appears to be 9.998 light-seconds away at t=19.998 seconds, and conclude that B was actually 9.998 light-seconds away at t=19.998-9.998=10 seconds in her frame.
You could also imagine that, to avoid the issue of light delays, A constructs a large network of rulers at rest in her frame and synchronized clocks fixed at regular intervals along the rulers, so that to assign coordinates to an event, she looks at the marking on the ruler that was right next to the event when it happened and the reading on the clock that was right next to the event as it happened. This is how Einstein originally imagined that inertial coordinate systems should be physically constructed in his 1905 paper. The tricky part is that each observer must decide how to "synchronize" different clocks in their system-they can't just synchronize them at a single location and then move them apart, because moving them apart will introduce time dilation. Instead, Einstein suggested that each observer synchronize his clocks using the assumption that light travels at the same speed in all directions, so that if you set off a flash at the midpoint of two clocks in your system, you call them "synchronized" if both clocks read the same time at the moment the light from the flash hits them. But this implies that different observers must have different definitions of simultaneity (they will disagree about whether two distant events happened 'at the same time'), since if I see your clocks moving in my frame, I will see the back clock moving towards the point where the flash was emitted and the front clock moving away from it, so if I assume light travels at the same speed in all directions in my frame I must conclude that the light will catch up with the back clock before it catches up with the front one.