The company surveyed 2,200 children about their online activities when it visited 45 schools across the country.
Company founder Joe Kenny said 17% of children in first class reported interacting with strangers online.
This should be of concern to parents and educators, he said.
Mr Kenny said parents were finding it more difficult to restrict their childrens access to the internet, with mobile phones and hand-held computers becoming more available and less expensive.
Many parents tell us that their children have apps or are using social-media sites without their consent, he said.
A growing number of primary school children are using tablets to access the internet, play games and use apps. The study showed 39% of pupils in first class use them instead of smart phones, laptops, consoles, or iPods.
Half of children in fifth class are allowed two hours screen time every day, while over 30% of children in first and second classes are spending between one and two hours a day in front of screens.
The study found that 26% of children in third class, 24% in fourth class, and 23% in fifth class said they had been cyberbullied or knew someone who had been a victim of online bullying.
Mr Kenny, an entrepreneur and a parent, said he established Zeeko in 2013 with backing from Enterprise Ireland and personal funding, concerned at the growing dominance of the internet in childrens lives.
Our advice to parents is that the best thing they can do is teach children how to navigate the internet safely, he said.
Cyber-psychology expert Dr Graacute;inne Kirwan said the internet had become such a part of daily life it would soon be impossible for parents to restrict access to it.
Parents and educators need to guide children on how to behave in online environments to reduce risk and to provide support for their problems and questions, she said.
This is no different from the guidance children receive in the offline world, where rules are well established for managing risky behaviours, for example, crossing the road, learning how to drive, or cycling in traffic.