There are definitely issues with finding time when one is already brilliantly busy. Perhaps there is a way to dedicate time specifically for such tasks. For example, at Gems Wellington, Qatar we have an HPL team with representatives from across the school who meet fortnightly. It could be something to consider if this is not already in place and this time could be dedicated to groups conducting research about particular topics. We also use Microsoft Teams and have a space where the team are able to contribute our opinions, reflections. resources and research. The same platform provides a further opportunity for the whole school to contribute in order for everyone to share best practice.
As far as developing a research culture in the workplace, I would recommend finding a few scholarly articles or books about fostering research cultures and around lifelong learning in the workplace.
It could be worth asking for volunteers who are willing to make the time to complete research and if necessary to present that information to either team leaders or the whole school during a professional development session. Alternatively, perhaps it is worth creating an online space where anyone can share any useful insights or articles as mentioned above.