There are several kinds of federal support for universities: direct support through Title III and similar grants for particular kinds of institutions (tribal institutions, Hispanic-serving institutions, etc.), indirect support through Title IV grants (i.e. financial aid to students) and research subventions. So it's not an easy question to answer. It's one thing to find out the total amounts expended in each annual federal budget, but I've never seen a single figure for all expenditures of all colleges and universities.
I'd recommend looking at institution-specific data, which are easy to find at the
IPEDS Data Center. Identify a series of peer institutions, then pull the revenue information from their financial reports and financial aid data from that report. These rates are going to vary widely -- a local HBCU may receive Title III funds, while a local research university receives many more thousands in federal research funding -- but if you keep to a group of similar institutions you can get a good handle on the problem.
And I'll second the motion to get help from your IR office; they live for this stuff.