I had 3 boys and 1 girl in scouting. After I sewed everything by hand for the first one I figured there had to be a better way! I use a zigzag stitch with either a clear nylon or polyester thread in the needle and a bobbin thread that matched the article I was sewing on (brown or green usually). I don't think a cheap modern machine will do it. Either buy a vintage black Singer for $30 (not too hard to figure out if it works first) or get someone in the troop to sew them for you. I used to do it for $1/patch, payment made to the troop fund. The dry cleaning companies charge $3-5 a patch!
Pam