There are a few ways to get involved in research on campus. One would be to check out the Student Involvement page from the Office of Research at UM. You could also visit or contact the Opportunity Hub to see what they may have available.
As for opportunities for research within the Department of Economics in particular, you have a couple of options. To work with a faculty member on their research, you would want to contact faculty directly about opportunities.
If you were interested in developing your own research, you could contact an Economics Professor directly about arranging an Independent Study course with them for Econ 499 credit (you can find some more information about Econ 499 here). You could also apply to take an Econ 495 seminar aimed at helping students develop skills and pursue independent research (topics vary by instructor, so please see the LSA Course Guide for topics offered in a given term).
There are also courses listed under "ALA" in the course guide for those interested in research, and UROP may have some summer opportunities.
You may find opportunities (or ideas) through contacting the Summer Research Opportunity Program (SROP).
You are of course always welcome to set up an appointment with an advisor to discuss other opportunities not have covered above.