Free subscription Out of Storage

If I am out of storage on my free subscription and someone tries to submit a new form, does that form go through and the oldest form in my file drop out?

Yes for your submission storage the most recent 500 will remain. If a new submission comes in it will go at the top and the bottom gets moved out. However, it is still possible to view the ones that are replaced when you upgrade to any paid plan.

Plans start at $19 USD per month.

https://www.jotform.com/pricing

Note that if you are over your monthly submission limit, your forms will not accept new submissions.