The fox door in Deltarune Chapter 4 is opened by finding the required foxes in that area first, then returning to the locked door and interacting with it once the count is complete.

What to do

  1. Progress through the statue/paper-wall area and collect the foxes hidden around the room.
  1. Look for foxes disguised as scenery, including a wandering fox, a fox behind a paper wall, and one revealed by moving a statue.
  1. After you have the required foxes, go back to the locked door near the back of the area and interact with it.
  1. The door should open once the requirement is met.

Useful clue

A guide for Chapter 4 says the main progression involves the rightmost door , while another item is tied to the leftmost door, which suggests the fox-door puzzle is part of a route split in that section.

Important note

The search results I found are a bit mixed between Chapter 4 and Chapter 5, so the safest read is that the “fox door” puzzle works by gathering hidden foxes in the room first, then returning to the locked door.

TL;DR

Find the hidden foxes in the area, then revisit the locked door and interact with it to open it.