Priests certainly existed before the law was given:
And Melchizedek king of Salem brought out bread and wine. (He was priest of God Most High.) (Gen 14:18)
Now Moses was keeping the flock of his father-in-law, Jethro, the priest of Midian, and he led his flock to the west side of the wilderness and came to Horeb, the mountain of God. (Exo 3:1)
So certainly the concept of priests was well understood when God said:
Now therefore, if you will indeed obey my voice and keep my covenant, you shall be my treasured possession among all peoples, for all the earth is mine; and you shall be to me a kingdom of priests and a holy nation. These are the words that you shall speak to the people of Israel.” (Exo 19:5 – 6)
And it appears that some of the Israelites served as priests at this time:
And Moses said to the LORD, “The people cannot come up to Mount Sinai, for you yourself warned us, saying, ‘Set limits around the mountain and consecrate it.'” And the LORD said to him, “Go down, and come up bringing Aaron with you. But do not let the priests and the people break through to come up to the LORD, lest he break out against them.” (Exo 19:23-24)
But we don’t really know what these priests did; they certainly didn’t have any formal role defined by a law of God that the Bible tells us about. As part of the law, God defined all this:
“Then bring near to you Aaron your brother, and his sons with him, from among the people of Israel, to serve me as priests–Aaron and Aaron’s sons, Nadab and Abihu, Eleazar and Ithamar. (Exo 28:1)
And then there’s more than a whole book (most of Leviticus and some of Numbers) about the various roles and responsibilities of the priests.