Peter Scott of Newport has seen empty shelves at the town's Waitrose branch where people have been buying things like toilet roll and flour.
In the early stages of the coronavirus crisis in the UK, supermarket chains had to put limits on certain items after people stockpiled packets of pasta, tins of food and packs of toilet paper.
Councillor Scott said: "There is absolutely no need to panic buy anything. Nothing has changed since the Prime Minister's broadcast.
"We are not short of anything. I would ask everyone to apply some common sense. No need to stock up."
The outgoing chief executive of Tesco, Dave Lewis, told Sky News that there is no need to panic buy this time around as supply chains have not been disrupted like they were when lockdown loomed.
Meanwhile, a director at the British Retail Consortium (BRC) has urged consumers to be considerate of others and “shop as you normally would”.
Director of food and sustainability at the BRC, Andrew Opie, said: “Supply chains are stronger than ever before and we do not anticipate any issues in the availability of food or other goods under a future lockdown.
“Nonetheless, we urge consumers to be considerate of others and shop as they normally would.”
He added that retailers have done an “excellent job” making sure customers have access to food throughout the pandemic, while installing perspex screens and other social distancing measures.
“As such, retail remains a safe space for consumers, even under future lockdowns,” he said.