Every time I try to create a site for a template, the sites list and template list (Manager of the websites templates) empty out and nothing is listed. I tried reinstalling JMS Multisites and the same thing happens again. Any idea how I can fix this issue? I cannot even retry, since the templates list defaults are no longer there. I need to reinstall in order to get them back and then it would just happen again.

Your question is not clear.

Do you mean the list of template in the front-end for the website creation from the front-end.
If this is from the front-end, the list is empty because you have forgotten to enter in the menu item the "group" value and to enter in the JMS templates the same group value.

The group fied in the JMS template is used to filter the list of templates that are displayed in the front-end.
So put for example in the JMS template rule the group = 'front-end' and put also exactly the same value in the menu item.

I suggest that you review the tutorial video 14.b, 14, 13.b and 13 that show how to create website from the front-end and watch what we put in the group field.

If your question is not related to the website creation from the front-end, please explain more in detail the scenario to reproduce it and give exact menu that you call with exact name. That will help following the scenario and understand the error or missing value.