I have constructed a stand here which holds the webcam at exactly the right distance for the QR code.

Constructed for the webcam of the model Aukey PC-LM3

Webcam is available here for only 5 CHF:
https://www.brack.ch/aukey-webcam-pc-lm3-1080p-1108405

You also need 4 screws M3x16mm or similar length.

CAD Model:
https://cad.onshape.com/documents/54ea080abbe53263ac344dda/w/997f6d1e3a3e5e0f81f0babb/e/c317549cf44f57f9f3f58cd5?renderMode=0&uiState=64f7871eb17723316471427e