I actually figured out what was wrong. It had to do with the fact that I’m using Gmail to run the email for the domain WP is installed on, and so you can’t use the standard php mail() function to send emails via Contact Form 7 or ANY other contact form program.
Here’s the solution:
1. Install the wp-smtp-mail plugin onto your WP site.
2. Follow the directions on https://www.callum-macdonald.com/code/wp-mail-smtp/ to configure the plugin properly (I used the Gmail/Google Apps settings contained on this page to properly configure the plugin for my domain)
3. Send yourself a test email using the plugin. When things are setup properly, the email actually sends and it (almost) instantaneously appears in your inbox. From there CF7 and other contact functions should work flawlessly.