Don’t donate to an organization that refuses to provide its Form 990 — it either doesn’t want your support or has something to hide. And in either case, it is likely breaking the law.
A wealth of information is available from an organization’s most recent IRS tax return (Form 990, for example). In addition to telling you a lot about how the organization spends the moneys entrusted to them, it will tell you about programs, people involved, cash reserves and debts.
Federal law requires tax-exempt organizations to make their Federal information (tax) returns publicly available. They can do so (i) by giving or mailing you a copy or (ii) by making them available on their own websites or on a site like guidestar.org. See Code of Federal Regulations 26 CFR Section 301.6104(d)-1 and 26 CFR Section 301.6104(d)-2 for the details.