I believe that what we are and what we think and feel is embodied in the structure of the brain and nervous system.
The brain and nervous system is the hardware - period. The software is our soul, which is the seat of the intellect, emotions, and will. Your PC's computer hardware is complex and designed to do a lot of things, but it will not run without the software drivers. Similarly, our human hardware - the brain and the nervous system - while being infinitely more complex than any man-made computer system - is still non-functional until it is animated by the soul. If you doubt this, look closely at the dead body in the casket the next time you attend a funeral. Is it capable of doing anything on its own? I saw my father breathing, talking, moving, thinking, etc. before he died. After the moment of death had passed, his dead body was doing none of those things. The body without the soul is dead.
So I believe that our sense of right and wrong is innate -- with details, customs, laws, manners, provided by culture...the underlying certainty that some things are right and some things are wrong, is inborn.
Oddly enough, your choice of words argues my own point of view rather than your own. Further, I would submit that deep down inside you know this truth and have unwittingly betrayed yourself. If morality is merely 'embodied in the structure of the brain and nervous system', then it is not "innate" or "inborn" - it is simply part of the hardware, and as such is not something we can properly call morality at all. You might just as well talk about the morals of your car's engine.
Morality is a word which speaks of principles or rules of conduct, and conformity to ideals of right human conduct. It involves knowing what is right and wrong, and choosing whether to do right or do wrong. These are not hardware concepts in the slightest. These concepts require intellect, emotions, and will. The hardware - your brain and nervous system - is built and wired to SUPPORT these things from a physical standpoint, but it does not follow that the brain itself IS those things. Your car does not know to make a left or right turn or go straight or backwards at an intersection, but it is designed so that when you, the driver, choose which of those actions you want to take and turn the wheel or hit the gas or brake or throw the car into reverse, it does the action, based on your input.
The awareness of basic right and wrong (morality) is indeed innate and inborn. In saying that, you have spoken correctly. But since morality is not the brain itself...then where did this awareness of Morality come from? It is obvious that none of us could have installed it ourselves. Therefore, an Outside Agency must have installed it into us. C.S. Lewis makes this argument quite convincingly in his book "Mere Christianity". The same Creator who made your hardware made your software also.
And your evidence for the separate hardware and software is...
With all due respect, this argument is premised on a fundamental misunderstanding of what you are talking about. There is no fundamental distinction between hardware and software; they are interchangeable and identical. Just because we choose to make that distinction based on how our hardware and software is typically designed does not make it meaningful in any type of theoretical sense. In other words, based on your argument the brain IS the soul, which while not what you intended, it is in fact the correct interpretation.
I don't fault you for your limited experience with this domain though; it is a very common and pedestrian misunderstanding. There are classes of universal computing machines in which the data, the program, and the machinery itself are all the exact same thing. This theoretical mathematical machinery looks and behaves a lot like the human brain in virtually every measurable aspect, so there are many people in the field who would state that the brain is a biological approximation of this class of machinery. It is ironic that so-called "neural networks" in computer science are essentially a failed model of universal computation, but a class of mathematically-derived machinery which claims no heritage from biology appears to be the correct analog.