Go to the General store and give a quarter to Mrs. Phelps to buy a dirty magazine.
Go to the police station. If Sheriff Dwayne is still there, leave and enter again.
Give the dirty magazine to Deputy Loomis and "enjoy" the cutscene.
Open the deputy's desk drawer and take the evidence room key, note, checkbook ledger, etc.
Use the evidence room key on the door to the far right.
Enter the room and get the sneakers, gas can and camera.
Enter the post office and give the gas can to Postmaster Boyle.
Use the pen in your inventory on the blank lodge application, then give the "filled out" application to the Sergeant at Arms.
BONUS: You can get a "get out of jail" card from Sheriff Dwayne.
This will let you kill any NPC you want, though you might softlock yourself if you kill the wrong NPC.
To do this, make a copy of the checkbook at the general store, and show Sheriff Dwayne the copy.
(If you show him the original, you'll get a fun game over animation.)