Can I make it so the jotform submits even if someone doesn't fill out the whole form?

Right now I only get completion if someone fills out the entire form and presses submit. 

Is there a way for me to still get the submission without them having to physically press submit? 

No, that's not possible. The form must be submitted by clicking its submit button before you will get submissions.

The only workaround I could think is to assign a unique session value for each user so you can check their data if they have filled out the form by checking the form URL with the session parameter.

Example: Let's assume you've sent the following URL to me then I filled out the fields without submitting it. When you access the form URL, you should see my unsubmitted data.

https://form.jotform.com/92032933548257?session=user1

Make sure to change the session value user1 for each user and it should be unique for everyone.