Gripe Day appears to be a day for the Chief of Berk to listen to the grievances of his people individually. Grievances appear not to need to be related to governance, but anything, such as Sven's complaint about the rambunctiousness of his lambs that he keeps in his home. The Chief then suggests a solution to the presented issue.
Grievances are heard one at a time in the Great Hall, with the Chief sitting on his throne. Typically there is a long single-file line going out of the building with people waiting for their turn to be heard. It is not mentioned how frequently Gripe Day occurs.