Basically the choice between a backpack and a shoulder bag will be determined by the mission president of your missionary's mission. Our missionary was informed in his booklet of general instructions that shoulder bags were considered more professional and strongly encouraged in his mission in SoCal.
Shoulder bags were extremely hard to find. We finally had success at the Missionary Mall in Provo, but they are not cheap. They run about $80, but with added bells and whistles, can run over $100. Later our missionary also found them at the MTC bookstore.
Backpacks are, of course, easier to find, but they need to be black or dark blue with no visible logos or insignias. Finding those, I think, would be harder.
Known as tracting bags, these bags are designed to carry the essentials for daily missionary work like missionary planners, pamphlets, pass-along cards, cameras, keys, etc. Some have room for rain gear and water bottles.
A tracting bag would make a nice birthday gift for a 19 year-old future missionary.