The video was sent to afternoon radio host Ben Fordham by listener Janet.
Janet's daughter Alisha filmed the incident outside Woolworths at Westfield Hurstville.
One woman in the video manages to buy five tins in one go, which she then packs in her trolley before rushing back in the store to buy more.
As the store is busy, the brazen actions of the frenzied shoppers go unnoticed by staff.
A disgusted shopper can be heard in the background describing the incident as embarrassing.
Janet told Fordham she called Woolworths' head office while the incident was taking place, asking for store management to intervene. The retail giant allegedly refused to take action.
"We have a two tin limit per customer transaction on baby formula in place due to supply constraints in the market," a Woolworths spokesperson told Daily Mail.
"Since we reintroduced the two tin limit, we've seen better availability of formula stock on our shelves."
Watch the shocking video below.