Web Hosting provider problem
After a frustrating lack of response or resolution from our current web host provider (aplus.net), I decided to begin transfer to a new hosting provider (ministryweb.org). After uploading our same web site (including Contact Form Solution Pack pages) to our new pre-prop site, the CAPTCHA security code works fine. See new pre-prop url at:
So this is clearly a problem related to our current (and soon former) web host provider. But for future reference, it would still be helpful to know what could cause this problem.