I spent a number of years working in public accounting in the DC area, almost exclusively with non-profit organizations on both the audit and tax side. Tax exempt organizations come under a fair amount of scrutiny, because of how valuable their tax exemption status is, and the government checks everything from executive compensation to programatic expenditures. Most have to get audited annually, and you can read their audited financial statements just like how you could read Apple's or Google's [1].
A key note about tax exempt organizations is that as soon as they start to earn income from activities that compete with typical for-profit organizations, they have to pay corporate taxes on that income. I saw some people implying that Amazon could become a subcompany of WaPo and avoid taxes, which, while technically is in the realm of possibility, would have 3 major consequences: firstly, the 501(c)3 non-profit parent organization can't make dividend-like disbursements to shareholders, so the money would be tied up in the non-profit, secondly, excessive compensation to employees could result in massive punitive taxes [2], and thirdly, the revenue that WaPo would receive would most likely be taxable unrelated business income. Bezos would be better off keeping Amazon separate and donating cash to WaPo -- at least then he'd be able to get a write off.
[0] irs.gov/charities-non-profits/charitable-organizations/exempt-purposes-internal-revenue-code-section-501c3
[1] There are several sites, but guidestar.org is the one I've mostly used.
[2] irs.gov/charities-non-profits/charitable-organizations/intermediate-sanctions-excess-benefit-transactions