It was easy to get the answer: 183 of the 1,066 who have enough data in IPEDS to calculate such things. That's 17%. For some, it's understandable: Berea College, Cooper Union, the College of the Ozarks, or Olin, for instance. They all have special populations and special funding models. But what about the others?
I arrayed the world of private, not-for-profits on a scatter chart, on top. The x-axis is calculated mean ACT scores (the average of the 25th and 75th percentile as reported in IPEDS). The y-axis is the draw rate, which is yield rate divided by admit rate. It's a power measure of market position, notwithstanding some very small non-selective institutions report bad data to IPEDS that makes their draw look high. (The average draw is about .7, for instance; but Harvard's is 14, and Stanford's is 10.5).
The bubbles are colored by a ratio: The numerator is the average amount of aid for aided students; the denominator is the average aid for all students enrolling. If everyone gets a scholarship, that ratio is 1:1; if fewer students get aid and the aid is a small amount, it's higher. I capped it at 3:1.
You can use the filters to look at a smaller group: Region, state, draw rates, religious affiliation, percent aided, or any variables in any combination. If you want to see the institutions where every single freshman gets a scholarship, you can simply look at the bar charts at bottom, where that information is displayed. They're sorted by IPEDS ID because different colleges often have the same name.
And note: I started with just bachelor's, master's, and doctoral/research institutions, but if you want to add other types, use the Carnegie filter.