What are some social/economic institutions that are used to solve the economic problem in a market economy?
The "economic problem" is comprised of three questions that all economies must answer. These are what to produce, how to produce it, and for whom to produce it. In a modern market economy, various institutions have been created to help answer these questions.
Universities are one institution used to help answer these questions. Universities typically conduct research and use that research to teach students how to answer the three questions above. Scientific research can help with the technical aspects of creating things. Research into business practices can help businesspeople know how to organize their businesses to produce at maximum efficiency.
A second example is a trade organization. Companies in similar businesses sometimes make organizations that can help them with their businesses. These organizations can share information on best practices or can (more often) help the industry as a whole lobby the government for policies that are more helpful to it.
Finally, you can even say that things like schools and churches help to solve the economic problem. These socialize us to have certain values. These values end up helping to determine the answers to some of the questions in the economic problem. For example, if churches teach us to be charitable, this helps us to answer the question of "for whom to produce."