Premise of the argument
We cannot conclusively say who made the first sacrifices and there is no direct evidence that God commanded it to Cain, Able, Adam or Eve, but the evidence is very heavy that Adam and Eve began the first sacrifices shortly after the fall and it was ordained by God and likely that He instructed them in it or even did the first one Himself.
It is definitely clear that sacrifice was instituted very early.
Abel and Cain apparently performed sacrifices, Abel bringing animal material and Cain bringing plant material. The ensuing rivalry between the two led to the infamous 'first murder'. It has been argued that Cain knew animal material and animal life was the requirement and his obvious 'cannot be wrong' attitude and jealousy was what led him to murder his own brother.
But were they the first to sacrifice? There are only a few people before them who might have been the first, assuming you take a literal view of Genesis.
One should think that Cain and Abel were told by someone that it was necessary, else why would they do it? In our modern times sacrifice has become synonymous with giving up something valued because it is a burden to actually kill and burn your own livestock when you could use it for food instead.
One interpretation says that it is possible that God Himself them to do so, considering they were also on the same plain speaking terms, demonstrated when God questions and subsequently curses Cain. But how should we interpret the curious statements in Genesis 3:21, which is just after God declared punishments for the fall?
The Lord God made garments of skin for Adam and his wife and clothed them.
God Himself made the garments for Adam and Eve. Depending on your views of whether there was death before the fall this may or may not be significant, but I am arguing for the literal interpretation of Genesis. So there was no death mentioned until this verse and it is implied that there would be no death if it were not for the fall, therefore, there is something significant about it.
So many have hypothesized that God Himself actually performed the very first sacrifice for Adam's and Eve's sins. This site argues that for example. God always leads by example, so Adam and Eve then mimicked God's righteous actions and began performing sacrifice. Whether you take this view or not it is indisputable that God approved of the animal's killing so that would leave only Adam or Eve to actually perform the sacrifice. It really does not matter, though, who did it first because just one generation later Cain and Able are sacrificing as if it is a regular thing, showing that it was already instituted and the most likely time was Genesis 3:21. Also, if God did this first sacrifice, He did not do any others, according to the text of the Bible, again leaving the task to Adam and Eve. I should note that you use the word 'instituted' but that implies religious rites are started my men. That is not biblical, especially to a literalist. To a literalist, God instituted all rites and ceremonies, it is just a matter of when He told mankind to practice them.
Asking if sacrifice developed naturally begs a question of why? Why would it because it is a terrible burden? Humans might feel guilty for wrong doing but I doubt to that extent. Also the evidence that God performed the first sacrifice immediately after the Fall indicates that it very much has to do with sin. Romans states plainly that the wages of sin is death and John 3:16 states plainly that we all perish because we need a sacrifice (which happens to be Jesus).
But for the sake of the texts we do have we should note that it was not until the reveling of the Law that frequency and reasons for sacrifice were known to us at least. This is more than 1000 years later. It is clear that there were thanksgiving reasons, as Noah did in after surviving the Flood, and it is implied that Cain and Able were performing a possibly annual or other time frame sacrifice as required by God. There are also a few examples of reasons for sacrifice in the story of Abraham. The most notable being told to do so by an angel of God when Abraham was told to sacrifice his own son Isaac. So there may have been very well understood reasons by the ancient persons for sacrifice that may have even been started with Adam and Eve shortly after the fall, but the only indication of those 'rules' becoming written is Leviticus with Moses. It is entirely possible that those sections in Leviticus were only parroting what had been done since the beginning.
So we cannot conclusively say that Adam and Eve were the first to make sacrifices and there is no direct evidence that God commanded it (except much later in the Law), but the evidence is very heavy that they did and it was ordained by God and likely that He instructed them in it or even did the first one Himself.