The manager claims as the woman was leaving the store she leaned into the doorway and breathed in her face before driving away honking her horn.
Staff then believe the woman called the store about 40 times following the incident.
The shop posted a picture of the woman to its Facebook page, informing her that staff had her details and the police had been informed.
"To the lady that came in not wearing a mask, was rude to our staff, deliberately breathed on a staff member then started ringing our phone number and hanging up non stop," the store wrote.
"Take note of these things: Caller ID (you're in the white pages and you live more than 5km away), you signed in the guest book, police."
Under current COVID-19 restrictions, people in Melbourne cannot travel more than 5km from their homes for shopping or exercise.
The only reason people can travel outside the 5km radius is for "work, medical care, primary and secondary education, childcare and care or compassionate reasons", according to the Department of Health and Human Services website.
Social media users were appalled by the woman's behaviour, calling for her to be fined.
"I'm really sorry you had to put up with that in your workplace, where you should feel safe and respected. I hope appropriate actions are taken so she won't do this anywhere else," one commenter wrote.
"The thing is, this pandemic has been shown to be deadly in all ages and backgrounds. It's also shown to be highly infectious, and people can often be asymptomatic. At the very least, this is assault. At the other extreme, potentially manslaughter. Shocking that someone with this little regard for humanity is part of our society," another person said.
Others branded her and "idiot, calling her behaviour "disgusting" and "disrespectful".
Multiple people branded her a "Karen".
"I bet her name is Karen," one person said.
The pet store confirmed the police had been contacted and were "going to act".
