Limit access to people with an account only

is there any way to restrict a form only for users with an account? We want a way to ensure that the person submitting the form is really who they say they are. 

You can do that by embedding your form into a secure page. For example, if you have a login page on your website - where people have an account with you, you could show the form on one of the pages behind the login.

If you are thinking that someone might still get the link of the form through the source code and want to prevent that as well, you could also add a widget such as: Get Form Page URL

This will basically always give you the page where the form is embedded as one of the values. If someone takes the form and opens it directly, the value would be different and you would immediately know how they submitted the form.

You could even add conditions to hide the submit button - or all fields, if the form is not embedded within the page.

What I really need to accomplish is to know for sure that the person who is filling out the form is actually who they say that they are. Is there anyway to restrict a form to only be signed by those who are sub-users of my account? We need this for HIPPA compliance. 

I think that you can use conditions to achieve it, when using conditions, you will be able to hide all the field in your form and show them only if the person who wants to fill it, is the right person.

I think that you should set a field for a secret password, and then share this password with the persons that you want to share your form.

This guide will help you doing that: http://www.jotform.com/help/316-How-to-show-or-hide-a-field-in-the-form-based-on-the-other-field-input-or-selection.