One software as service company I know of is HyperOffice. Back in the Internet boom they were trying to be an MS-Office competitor; after the bust they settled in and found a niche serving small companies and organizations without a technical staff -- similar to how you describe your organization.
http://www.hyperoffice.com/

They have some case studies on their non-profit clients:
http://www.hyperoffice.com/hypermain/non_profit.cfm

A disclaimer - I know one of the principals in the company. But as an IT executive with a large company who looks at software for a living, I can also tell you that their suite is quite nice. Pricing is published on their website; non-profits get a discount.