After you have calmed Divine Beast Vah Ruta, you can talk to Tumbo, a young Zora child, during the day to receive this quest. Tumbo is trying to compensate for some family gone missing and asks you to hunt for five Hot-Footed Frogs to donate to his cause.
This being a basic fetch quest, there is no particular location you need to go to for some frog hunting. Hot-Footed Frogs don't need rain, although they certainly do become more common. The frogs can be mostly found around the shores of shallow lakes, ponds, and puddles. They will scatter once they hear you, so you will have to run after them or increase your stealth.
If you have upgraded your Sheikah Sensor by completed the sidequest Slated For Upgrades, you can make this quest less tedious by tuning your sensor to detect Hot-Footed Frogs. Learn how to upgrade your Sheikah Slate here. Once you have found five, give them to Tumbo and as your reward, he will give you an Armoranth.