I assume you're using Portal and you do want to use Forms for all other functionalities (builder, notifier, etc,)
So probably easiest to use an Event Handler
docs.kentico.com/k12/.../handling-global-events
And use the BizFormItemEvents
docs.kentico.com/.../reference-global-system-events
To do something custom with the data after every insert.
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Alternative could be to create a custom Web Part that would handle the whole form by itself, but that might be a lot more work...