J
It looks like you can clear the Pokémon from one outbreak even if the outbreak has ended. From this https://www.ign.com/wikis/pokemon-legends-arceus/Massive_Mass_Outbreaks#How_to_Make_Massive_Mass_Outbreaks_Spawn :
However, if the Massive Mass Outbreak disappears while you are in the middle of investigating an outbreak, those particular Pokemon will remain. Clear them, and it will still count, even if the storm had dispersed!
This might suggest a strategy of trying to find 3 of them before it stops raining, even if you haven't finished all three.
If you aren't on the same map, though, or if you aren't in the middle of the third outbreak of that particular map, then you will need to respawn it by heading back to Jubilife Village. You'll get this message from Mai:
When you get back to Jubilife Village, if the rainstorm is gone, you'll see no Massive Mass Outbreak; a regular Mass Outbreak doesn't count.
The next time there is a rainstorm or blizzard, there will be another Massive Mass Outbreak. The easiest way to force this, rather than going back and forth frequently, is to force the weather first. This can be done by repeatedly resting - I usually go on a cycle Morning -> Midday -> Evening -> Morning (repeat), but it's possible this isn't the most efficient. Once you wake up to a rainstorm (the cloud-lightning bolt icon), you should go back to Jubilife Village immediately, and then turn around and come back; as long as the weather doesn't change in the short time for the trip (which I haven't notice it do), there will then be a Massive Mass Outbreak in the location you were in.
Complete the outbreak collection and you'll be good to go on the quest.