Parents fear making kids sit transfer tests in grammar school halls could put their health in danger.
Naomi McBurney began a campaign for them to be allowed to take the exam at their own primary schools before Covid-19 hit. But she said the pandemic and the the restrictions made it unsafe for children to "burst their bubbles" and sit the exam in large groups.
On Thursday the Health Minister announced new limits on social gatherings. He also said he expects there will be Covid cases in schools when they reopen.
Mrs McBurney's petition has topped 6,500 signatures.
She added: "This is imperative now, we've only a few months left and it's [the pandemic] getting worse not better.
"These private providers are saying it's OK to bring hundreds, thousands of primary school students together who wouldn't normally mix together."
But one of the test providers, the Association for Quality Education (AQE), said it would be impossible for children to take the test in their own school unless it was allowed at every campus, but there "no prospect of this happening".
The Department of Education said it was for individual schools to organise procedures with the testing organisations.