This is an excellent question. Karma is well accepted in Buddhism, but not the existence of the soul. This inconsistency raises the question of remnant reaction. Consider the following argument.
From the system of karma we know that a person who does the action will be the same person who will experience the reaction, not any one else. Suppose a person dies, then what will happen to the inexperienced remnant reactions? It can't just go away, can it? No, and we know from system of karma that the same person will born again to experience the pending reactions. So here is the question, what is common in both the persons that you say the new born one is the rebirth of the old deceased one? If the two persons are identical then there has to be something common in them, isn't it? So what is that fundamental unit that is common to both? It can't be the body, nor can it be the mind (as memory, tendencies, qualities, etc. are different). So what is the common element? The answer is soul. I have to keep it brief so can not go elaborate more but the short answer is soul exists.
Why did not Buddha acknowledge the existence of soul?
The mission and objective of Buddha was to ward of suffering of the people which is primarily caused by attachment. So He didn't acknowledge the existence of soul. But just because He didn't acknowledge its existence, it doesn't necessarily mean it doesn't exist. He didn't acknowledge the existence of soul because it will create attachment to an eternal self. And where there is attachment suffering will undoubtedly follow. During His time in India people were doing Vedic sacrifices, yanjya, rituals, etc. to place there self in heaven etc. As they believed in the eternal soul, they were attached to it and hence putting effort to place it in heaven and higher realms by animal sacrifices in yanjya and rituals. So He didn't acknowledge the existence of soul to free followers from attachment towards an eternal self.
However, if you go into more deep then you will find that the soul exists in such a way that its existence can be safely overlooked. So even if you deny the existence of the individual soul you will not be completely wrong.