Well this comes as much as a surprise as it doesn't, Bozai has specialized dialogue for when he can't give you snow boots due to full armor inventory : r/Breath_of_the_Wild
Wii U - The Legend of Zelda: Breath of the Wild - Snow Boots - The Models Resource
BOTW NPC DIALOGUE - Bozai the boot guy - Part 3 And here's the finale...
How to Get the Snow Boots | Hyrule Warriors: Age of Calamity|Game8
Snow Boots - The Legend of Zelda: Breath of the Wild Wiki Guide - IGN
Snow Boots Armor Guide [The Legend of Zelda: Breath of the Wild] - SAMURAI GAMERS
HOW TO GET THE SNOW BOOTS AND SAND BOOTS!! | Legend of Zelda: Breath of the Wild Guide - YouTube
Zelda: Breath of the Wild guide: The Forgotten Sword side quest walkthrough (How to get the snow boots) - Polygon
Zelda: Breath of the Wild guide: The Forgotten Sword side quest walkthrough (How to get the snow boots) - Polygon
HOW TO GET SAND BOOTS and SNOW BOOTS STEP BY STEP - Zelda: BotW - YouTube
How to Get Sand & Snow Boots ! The Eighth Heroine & The Forgotten Sword Side Quest - YouTube