One supermarket worker in Cardiff said work had been "crazy" and it had "affected my mental health".
Shop workers' union Usdaw said retail staff "play an essential role in getting the country through this crisis" and needed support.
It also called for precautions to protect them from catching the virus at work.
A worker at a Co-op store in Cardiff, who did not want to be named, described receiving racist abuse from a customer.
"He was shouting that it was a virus caused by the Chinese and that he had the right to freedom of speech," they said.
A Tesco employee from Cardiff, who also did not want to be identified, said: "Working here this week has really affected my mental health.
"People have been so selfish. It's just been crazy here with all the panic-buying.
"We've seen the worst behaviour this week, it's just been insane."
Usdaw said its members were dealing with extremely high demand and coming into contact with hundreds of people every day.
Usdaw's general secretary, Paddy Lillis, said: "We understand this is a stressful time and remind customers that shop workers deserve respect and that no level of abuse is ever acceptable.
"It should never be a part of the job."
A Co-op spokesman said: "We have taken immediate steps to help our front-line colleagues follow all the precautions advised by NHS and Public Health and we've built in additional working hours for store colleagues to undertake more frequent hand washing throughout the day.
"Additional hand sanitiser, antibacterial wipes and soap have been provided to all stores and they are all following upweighted, rigorous cleaning protocols of the store and all fixtures."
What are other supermarkets doing to protect people?
Aldi said it was taking steps to improve product availability to protect people against infection.
Store hours will shorten to close at 20:00 from this Friday, 20 March, until further notice. Sunday hours are unchanged.
Clear screens will be installed at checkouts to protect colleagues and customers from Monday, 23 March.
The upmarket store said it planned to increase security so there would be a guard at all branches.
The first hour of business is now dedicated to elderly and vulnerable shoppers.
It said it had a £1m fund to help those in need in local communities.
Tesco is limiting customers to three of any product, and only two of toilet rolls and paracetamol.
It said it will introduce distancing measures at checkouts to reduce infection risk.
It's also introducing a shopping hour on Sunday morning in large stores for its staff and NHS workers.
It said people can buy three of any grocery product and two of more popular items like toilet paper, soap and long-life milk.
It has given over its first hour of opening in stores, apart from its Local shops, to elderly and vulnerable customers.
Deeside based Iceland has been opening an hour early for elderly and vulnerable people.