You can find a Zora spear upstairs from the trident workshop, in the kiddy pools. Use Magnesis to find it. Note that every Zora weapon can be found laying around the Domain, with the bow being in the water below it.
You can find a Goron cobble breaker in the cave-in in the northern mines. I'm getting some unconfirmed reports that it's in the city, on the upper layers in a house in the northeast portion of the city. You can also find a goron spear through a quest up the mountain from Goron City.
You can find a Ruto bow right in front of the Ruto who forges the champion bow.
You can find a Gerudo scimitar on top of the boxes to the left of the entrance of Gerudo city. The shield is the hardest base item required for champion weapons to find, being dropped from a lizalfos in a small camp between Gerudo canyon and the bazaar. You can find a bow and spear back in Gerudo city, with the bow in the training yard and spear in the room next to Patricia's courtyard.
This should give you easy access to all champion weapons, along with any emergency weapons in case you somehow break all your weapons or display them in your home.