The city is on heightened alert level three after the confirmation of at least four COVID-19 cases.
Efeso Collins said the panic buying coincided with benefit payments, and some families trying to buy necessities missed out to people bulk buying.
He said many families were already struggling with a lot losing hours and jobs earlier in the pandemic.
"We're the ones who go in and we can only afford one tin of milk powder, and this is what we would describe as staples to get our families through the week.
"This is what families are relying on for the week. So if we're going in and there's toilet paper that's flying off the shelves and the milk's gone, then it's only gonna set a pattern of anxiety for our families," Collins said.
The Manukau Councillor said the Pacific community in the area was particularly vulnerable, with many being low-paid essential workers
Meanwhile Collins also called for the family who had tested positive for COVID-19 to be supported.
He said he had spoken to people who knew the family well, and understood they were embarrassed at their situation.
Collins said he and others in the South Auckland community were offering them whatever support they needed.
"This is them almost carrying a sense of guilt that they're the ones who have coronavirus at the moment. I think they've done everything right, good on them for getting tested, for presenting.
"The last thing we want is for these vulnerable families to feel it's their fault that we're now at level three."
RNZ
