One of the best ways to gauge a private foundation's financial capacity is to look at the organization's IRS Form 990 (nonprofit tax return). This form lists all the assets, major grants awarded, and contact information of its board members. The latter information is useful for when you want to approach a decision maker to champion your funding request.
By looking at the total number of grants awarded, you can get an idea of the funder’s financial capacity, which helps you determine your chances of receiving an award. These details can help you determine whether a grant opportunity is a go or a no-go, saving your organization or department significant time (= money).
Most Form 990s for nonprofits and grant makers can be found at GuideStar. You can also request Form 990s from the funder itself or from the IRS.