The Weapon Connoisseur
Nebb is available running around the middle of town roughly between noon and 7 p.m. After you have started the quest, when you need to find him again, you can set this quest as the active one in the Adventure Log.
Talk to Nebb to start the quest.
Tip: If you have done the Sunshroom Sensing Side Quest, have Sensor+, and have taken a picture of the requested weapon, you can set the weapon as your Sensor+ target for help in finding one. If you have not taken a picture of the weapon, you can talk to Symin at the Hateno Ancient Tech Lab and purchase equipment pictures for your compendium (at 100 rupees each, getting a random picture each time, can buy in bulk if desired) until you have a picture for your next objective. You can then set it as your Sensor+ target for help in finding one.
Each of these weapons likely have multiple locations. The locations listed below are simply suggestions.
If you are already carrying the requested weapon, speak to Nebb again to show him the weapon. It does not need to be equipped.
Objective 1: Traveler's Sword
A Traveler's Sword can be found on the path just east of Inogo Bridge. (If using Sensor+, this is #204 in the Compendium.) He will give you a red rupee.
Objective 2: Fire Rod
A Fire Rod can be found on an enemy close to the Wasteland Tower or the Woodland Tower. (If using Sensor+, this is #233 in the Compendium.) Show it to Nebb for a red rupee.
Objective 3: Moblin Club
Tip: The two Moblins at the north end of Digdogg Suspension Bridge carry Moblin Clubs near the beginning of the game.
Objective 4: Duplex Bow
Duplex Bows can be found on most Yiga Footsoldiers, especially those outside their hideout. (If using Sensor+, this is #343 in the Compendium.) Show it to Nebb for a purple rupee.
Objective 5: Windcleaver
Found on Yiga Blademasters. They are available in various places (Hyrule Field, Akkala Highlands in particular) but become more available after clearing the Yiga Clan Hideout. (If using Sensor+, this is #279 in the Compendium.) Show it to Nebb for a silver rupee.
Objective 6: Ancient Battle Axe+
The Ancient Battle Axe+ drops off Guardian Scout III, which can be found at many of the "Modest Test of Strength" shrines. As long as a blood moon has passed, you can revisit and kill the enemy again even if you have already beaten the shrine. (If using Sensor+, this is #274 in the Compendium.) Show it to Nebb for a silver rupee.
Objective 7: Frostspear
If you have not done it yet, you can get a Frostspear at the shrine in the Spring of Wisdom (Jitan Sa'mi Shrine). You can also get it off a Stalnox in the Lake region, or from the Coliseum Ruins. (If using Sensor+, this is #302 in the Compendium.) Show it to Nebb for a gold rupee.
Objective 8: Ancient Short Sword
Tip - The Minor Test of Strength Shrines commonly contain Guardian Scouts using the Ancient Short Sword.