What Windows app lets me sort photos into folders with keyboard shortcuts?

XnView may be a good fit. It can display the image and lets you choose a destination folder when using its move command, with Alt+M available as a shortcut. It may not be exactly one key per folder, but it should be much faster than manually switching windows.

Another option is an AutoHotkey script. You could have it detect the currently open image, then assign several keys to copy or move that file into specific folders. That gives you a true one-key workflow once the script is configured.

Total Commander is worth checking out if you want a file manager with keyboard-driven workflows. Pairing it with its image viewer or a preview pane can make moving and copying files between folders very quick, although it may take some setup to match your preferred keybinds.