When our son was called to his "biking" mission, he received the following information in the booklet he received with his call . Under the heading "General Instruction Checklist: Miscellaneous," it read, "If you are assigned to an area in your mission where bicycles are used, you will be expected to purchase both a bicycle and a helmet. Use of a helmet is mandatory. You and your family will need to provide the money to purchase a bicycle and helmet when needed. The current cost of a bicycle and helmet is US $350. It may not be practical to ship these items from your home."
Not very long on specifics as you can see.
The first thing we did was to contact the mission home where our son was going to be serving and see if there was a vendor they recommended. Fortunately for us, we were directed to Bob's Bikes.
Located in Southern California but serving missions nationwide, Bob's Bikes is an approved vendor for The Church of Jesus Christ of Latter-day Saints. The Kona bikes advertised on his website range in price from $389 to $599, though other models are available. All the bikes are made with high quality aluminum frames with replaceable rear dropouts, shimano drivetrains, and front suspension. The price includes an accessory package that includes a kickstand, helmet, front and rear light kit, cable or U-lock, multi-tool, water bottle and cage. If your missionary will be serving in a SoCal mission, Bob will assemble the bike and deliver it to their first transfer meeting. He will also maintain the bicycle during their mission. For more information or recommendations, Bob can also be reached at (619) 507-7424 or by email at firstname.lastname@example.org.
Bob was actually able to furnish us with a refurbished bicycle that a former missionary had left with him on consignment. Bob replaced some of the parts that had excessive wear, and then he delivered it to our son on his arrival. The total for the refurbished bike and the accessory package was $250.
My recommendation would be to contact the mission office first and ask a lot of questions, but if they do not have a preferred vendor, look for a bike with an aluminum frame that comes assembled. And do not forget the helmet!