Hi @jeffreygaudette,
Thanks for posting on the forum.
This type of issue mostly caused because of JavaScript error message generated by any plugin/theme that you are using along with this plugin. It happens because Appointments plugin uses AJAX and the functionality of it gets affected due to JavaScript error message.
Can you check the JavaScript console for errors?
In Chrome go to “View -> Developer -> JavaScript Console” and in Firefox you can go to “Tools -> Web Developer -> Console”
Find information about how to open JavaScript browser console :
https://webmasters.stackexchange.com/questions/8525/how-to-open-the-javascript-console-in-different-browsers
https://codex.www.ads-software.com/Using_Your_Browser_to_Diagnose_JavaScript_Errors
Let me know what JavaScript errors if any you are getting when you face this issue and which plugin/theme is causing it.
You can also just try on your development site using default WordPress theme like Twenty Fifteen and deactivating all other plugins as displayed in the following flowchart to know which theme/plugin is conflicting if any.
https://premium.wpmudev.org/wp-content/uploads/2013/12/Support-Process-Support-Process.png
Best Regards,
Vinod Dalvi