The proprietor of The Legend of Zelda: Breath of the Wild inventory upgrades, the giant Korok named Hestu, won't trade with Link if he hasn't got any clothes on. Link cannot be entirely naked, he just strips down to his underwear, an outfit (or lack thereof) that elicits unique responses from NPCs. This includes the jovial, dancing Hestu, who isn't bothered in the slightest by Link's risqué appearance, and is in fact quite elated by the interaction.
BOTW's expansive version of Hyrule has scattered a whopping 900 Korok Seeds for players to collect. Not every one is required to get all inventory upgrades - just 441 - but BOTW's final Korok Seed reward is a thoughtful gift from Hestu for all the trouble. Hestu is found in various locations around Hyrule, but can eventually be found at the heart of Korok Forest next to the Great Deku Tree. Every time Link returns with the proper number of Korok Seeds for an upgrade, Hestu sings, dances, and shakes his maracas to expand an inventory of the player's choosing.
Hestu's normal dance won't occur if players show up with no clothes equipped. Instead of being offended at Link's lack of clothing, Hestu is so excited he won't upgrade the player's inventory. He simply acknowledges Link's nakedness, comments that he too doesn't wear any clothes, and gives a short song and dance to celebrate how the two are "united in exposure." Players engaged in a strict armor-less run won't need to worry about getting BOTW's hardest-to-find Korok Seeds, since Hestu won't put them to use anyway.
Although most NPCs will just react with their canned response to Link engaging them in conversation, some will have some special words if he shows up in his under garments. Robbie and Purah, researchers in the Ancient Tech Labs, are largely unfazed by Link showing up without clothes on, both remarking on the scars he bears from the outbreak of the Second Great Calamity 100 years ago. Bolson and some of the other workers at Bolson Construction will comment on Link's rather trim physique. Some deft textile merchants will take the opportunity to make a situational sales pitch, commenting how their store is just the place for someone without clothes.
Aside from Hestu, Paya's reaction is probably the most entertaining. Paya is the granddaughter of BOTW's incarnation of The Legend of Zelda's Impa, and seems to have quite the crush on Link. She responds as most people probably should to the situation, by insisting Link cover himself while in public or in other people's homes. Paya is right in saying such behavior is unbecoming of a hero like Link, and the good citizens of Breath of the Wild's Hyrule would probably appreciate if he put some clothes on, but at least Hestu gets a kick out of the situation.
