Racial diversity is pretty straightforward - just look at CDS for each college, or use a site that uses CDS data, like the government’s College Navigator. Here is Yale’s racial breakdown: https://nces.ed.gov/collegenavigator/?q=yale&s=all&id=130794#enrolmt
For socioeconomic data, there’s Pell grant %, % receiving need-based aid data. Some of that is also at College Nav - https://nces.ed.gov/collegenavigator/?q=yale&s=all&id=130794#finaid
Pretty much every college in the country’s data is there.
Percentages don’t tell the whole story of course. But it’s one angle that’s fairly easy to start with.