How and why does reciprocity still play a vital role in a modern market economy like the USA?
I assume that you are talking about reciprocity of the sort that is practiced by "primitive" people. This is the sort of thing where people give gifts to one another as a way of transferring goods and services to one another. In this sort of a system, people tend to give one another things they need in the expectation that they will themselves will be able to get things when needed.
This still plays a role in modern economies because even modern economies are often based on personal relationships. You will have things like companies having golf outings and inviting suppliers and customers to come along. By giving these sorts of gifts to their suppliers and customers, the companies hope to build up good will with those people. They hope to be able to call on that good will some time in the future when they have a need.
Because personal relationships are still important, reciprocity plays a part even in a modern market economy.