Can forms be downloaded to a data system, and not connected to a website?

Yes, you can download the form's full source code. Here's the guide: How-to-get-the-Full-Source-Code-of-your-Form. Please take note that the source code contains the HTML, CSS and Javascript files of the form. If your data system supports HTML embed, then it should work. 

If you are planning to send the form submissions to an external database, then it is possible as well, please check these guides:
How-to-send-Submissions-to-Your-MySQL-Database-Using-PHP
How-to-Setup-a-Webhook-with-JotForm

You may also want to check our API Documentation. Here's the link: https://api.jotform.com/docs/.

Hope that helps. If you have any questions, let us know. Thank you.