I'm not sure how exactly the circulatory system developed in our ancestors, but I will add that simpler circulatory systems do exist in animals today, including the open circulatory systems that are found in, e.g., arthropods. In these animals, blood is pumped across an body cavity by a heart, before flowing back through the fluid-filled body cavity itself, outside of any blood vessels. It is definitely possible to have "partial" circulatory system that functions fine while lacking some of the components present in humans.

Similarly, simpler versions of, e.g, eyes have been observed in nature with structures that are thought to be analogous to those of our early ancestors when eyes were first developing.