In the ACT, where I am, the Act states: “The executive committee of an owners corporation may appoint a member of the corporation to fill a casual vacancy on the committee until the next annual general meeting.”
‘May’ means the EC could carry on without replacing. The EC could find someone willing by whatever means it likes, which could be conducting a vote of the OC, but it is not obliged to do so.
In our OC, when an EC member resigned part way through the year recently, the remaining EC put a short item in our newsletter. We thanked the person who had resigned, we pointed out that we traditionally try to encourage someone from each of 5 zones in our large site to nominate to be on the EC and we pointed out that we had no-one from one of those zones. Consequently, we would encourage anyone who would like to see what it is like to be on the EC to let us know, particularly if they were from the ‘unrepresented’ zone. We pointed out that this was an ideal opportunity for someone to see what it would be like to be on the EC without committing to a whole year. We did not get any expression of interest so we carried on with the remaining EC members for the rest of the year.
As for ‘Tricky Rikky’, I think you could point out that when a time period is not specified the reasonable interpretation is that the thing would be done promptly without unnecessary delay.