NSF does actually release those numbers as well. There are full tables here: [NSF Baccalaureate Origins of S&E Doctoral Students](nsf.gov - NCSES Baccalaureate Origins of U.S.-trained S&E Doctorate Recipients - US National Science Foundation (NSF)) They rank schools by institutional-yield ratio, which is the number of PhD recipients for that school per every 100 students the school graduated 9 years earlier. So for example, for every 100 Caltech grads that graduated with their bachelor’s between 1993 and 2002, about 35 have completed a PhD in a science & engineering field (which, for the NSF, includes the social and behavioral sciences, science and math education, and health sciences fields).
Scroll about 1/3 of the way down the page. Unsurprisingly, small LACs do much better here - Caltech tops the list, followed by Harvey Mudd, MIT, Reed, Swarthmore, Carleton, Grinnell, Rice, University of Chicago and Princeton to round out the top 10.