You can find the polygon data on OpenStreetMaps with the following tags
https://wiki.openstreetmap.org/wiki/Tag:amenity=university
and perhaps also
https://wiki.openstreetmap.org/wiki/Tag:amenity=college
The query would look like this
[out:json][timeout:25];
(
node["amenity"="university"]({{bbox}});
way["amenity"="university"]({{bbox}});
relation["amenity"="university"]({{bbox}});
);
out body;
>;
out skel qt;
Which you can paste into http://overpass-turbo.eu/ and run.

Zoom out and run again, to get all universities in the map view.
Each tagged object will have a multipolygon like this:

Click the data tab to see the data. Each object has all the geodata, as well as metadata like this:
"tags": {
"access": "yes",
"amenity": "university",
"boundary": "protected_area",
"landuse": "forest",
"leisure": "nature_reserve",
"name": "Charles Lathrop Pack Demonstration Forest",
"operator": "State University of New York College of Environmental Science and Forestry",
"operator:wikidata": "Q7603610",
"operator:wikipedia": "en:State University of New York College of Environmental Science and Forestry",
"phone": "+1 518 623 9679",
"protect_class": "6",
"protection_object": "education",
"site_ownership": "state",
"type": "multipolygon",
"website": "http://www.esf.edu/campuses/eec/"
}
Search this site or OSM forums for more details. You can also download the entire US (or world, see geofabrik) and then use osmfilter to keep the tags you want, and then use ogr2ogr to convert OSM formats to anything else (SQL, GeoJSON, etc). Here's an answer with the steps written out.