Apparently, there's no need to wear a padded costume when the dangerous supervillain equipment you're wielding has built-in protective features. Pretty much every cool and terrifying gadget Vulture's gang has access to is repurposed from gear that appeared in MCU movies that came before "Spider-Man: Homecoming," with some brand new extra features courtesy of Phineas "The Tinkerer" Mason (Michael Chernus). While the vast majority of their equipment seems to be from the aftermath of the climactic Battle of New York in "The Avengers," Shocker's gauntlet is actually an upgraded, electrified part of the suit Crossbones (Frank Grillo) wears to fight Steve Rogers (Chris Evans) in "Captain America: Civil War."
If this sounds like the kind of weapon that could potentially take your arm off, well, it absolutely is. Fortunately, the Tinkerer thought ahead and built in a "power dampener," which allows the Shocker to use the gauntlet without any danger to himself. Seeing as this dampener provides the same protective effect the Shocker costume does in the comics, it's only natural that the MCU Shocker doesn't need a special suit and is free to fight Spider-Man (Tom Holland) in his street clothes.