There are a number of opportunities to get your law-related papers published in undergraduate journals. See the list below for information on the journals and their respective deadlines. You'll notice that several of the Journals are published by other colleges and universities. However, they are only included below if they accept papers from undergraduates at other institutions.
Critical Theory and Social Justice Journal of Undergraduate Research (Occidental College)
The Dialectics: Journal of Law, Leadership, and Society (Penn State University)