Sorry, my bad. You are correct, it is set up to send alert.
Would this be the setting to allow multiple times? I am assuming that because 'only once' is selected, this is the reason it's happening?

This should be set to "every time", if you want to receive notification every time a person fills out the form. 🙂
I think you may not need this for this campaign in particular, but just FYR: there's an another option in this dialog to let people filling out the form run through this campaign "once every x hours/days/months , etc." - example, if you choose once every 2 days, the qualifying person will flow through the campaign only once every 2 days - even if the person qualifies for the campaign more than once in this timeframe unlike the "every time" option, where-in the person flows through the campaign every time they qualify for it. Hope you find this helpful, happy to answer any questions you have in this regard.