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Following up on this -- it looks like the site was hit on April 1st with some hacking, which seems to have included an altered robots.txt file. All replaced now and everything seems fine. Thanks for the advice!

You do have to go into your Client Area and download the Pages app and install on your server to access the options. I remember having this same issue. 🙂
In your Client Area, click the "Download Invision Community" button. Then, download the package. Only upload applications/cms folder, then go to the Applications page in your Admin CP. You will see Pages in Applications currently not installed section.

Hi! Thanks for all the suggestions. The user did everything, suggested and actually reinstalled browsers. They did change to the default theme (which took some explaining, because they were sure it was broken since it was just plain white with no design). Starting last night, however, everything seems to be working fine. Not sure what it was causing the issue, but thanks for the help. Something worked!

Is it possible that you have the board where this post was posted set to no adding to post count?

Okay, after some big digging around here, I've found the issue! While it is a theme issue, this has to be something that blipped in the update, as the issue is in a place that has never been modified. To fix it, I had to create a new theme using the default settings. 
Open new default theme and navigate to core/front/profile/profile. Copy entire text for that file. Paste into corresponding core/front/profile/profile area in the theme that needs fixing. Hit save. Worked perfectly, even though my original was unaltered.

Posting the answer on this, just in case someone else has this problem.
The response to the support ticket was:
The only current spam mitigation we have on the contact form is the CAPTCHA. You can try a different CAPTCHA if you think bots are getting past the current CAPTCHA you're using.
So, there is no way to filter e-mail addresses.