Submissions are counted on a per-Form basis. For example, on the Manager plan, you are allotted 10,000 Submissions on a Form which means you may never have more than the allotted submissions on a single Form at any given moment. At any point, you may export and delete submissions to free-up space for new Submissions to come in.
If you do get close to your submission limit, you will receive an email notification warning you that you are approaching your submission limit and you will receive another warning email once that limit has been reached.
Forms submitted after the submission limit has been exceeded will not be lost, rather they will be emailed to you instead of being stored in the Formstack database. Also, integrations will continue to run when the Form is submitted in the event that you exceed the submission limit.