The offices of institutional research at both schools will have this data. However, just the average of the whole incoming class does not tell the whole story. UM has a higher white % and a higher out-of-state % of incoming freshman, both of which will contribute to a higher average ACT score. If I had the time today and could find what I need, I'd love to do a comparison of incoming-freshman ACT scores by in-state vs. out-of-state and by racial/ethnic group.
MSU IR Main Page: http://www.ir.msstate.edu/
UM IR Main Page: http://irep.olemiss.edu/
MSU IR Office ACT Scores: http://www.ir.msstate.edu/enroll_ftf_act_avg_gender_race_fa10_fa16.pdf
UM IR Office Mini-Fact Book: http://irep.olemiss.edu/wp-content/uploads/sites/98/2016/12/Mini-Fact-Book-2016-2017.pdf