He is on a kick now where he says you can do older, retired merit badges. He wants to do one from the early 1900's! And find someone to make a badge for him to wear.
If you don't have anything around the house, a Ruger 10/22 is a great starting point. Leave the iron sights on it as the merit badge requires shooting with the stock open sights. A box of 500 rounds will cost you about $10 at most stores. Don't forget shooting glasses and ear protection. Even the lowly .22LR produces enough of a supersonic "crack" to damage hearing over a prolonged period.
See if you have an indoor range in your area. The required distance for the merit badget is 50 feet. Most indoor ranges are set up with 15 or 25 yard lanes with a target trolley to carry the target downrange. The 15 yard is just short of the requirement. Take a measuring tape and have the range officer call a halt so you can put some masking tape on the floor at the 50 foot point in the 25 yard lane.