They are being employed by the NCR force as a mean to find more NCR citizens in the area, so they can provide them with food and water. Each missionary is located in the squatter camps you are originally sent to investigate, they are the quest markers.
G.I. Blues: The King gives the player the optional objective to investigate the NCR squatters in Freeside to find out more about the people who attacked his men. The player can ignore that and go directly to Julie Farkas who will send you to talk to her friend, Major Elizabeth Kieran who is in charge of the food supplies at the Ruined store.
Either will ask the player if they are a citizen of the NCR. If the player answers "yes", the player is quizzed with three questions to prove that they are an NCR Citizen.
Question 1 - Who is the most popular NCR President of all time? President Tandi.
Question 2 - What was the original capital of the NCR? Shady Sands.
Question 3 - What animal is on the NCR Flag? Two-Headed Bear.
This will give you access to Missionary's note, which reads cryptically "Hope is the key." Hope is actually the password you give to the two NCR guards outside the ruined store they use to supply the NCR Freeside members with food and water. This note can also be obtained by killing the missionaries.
The quiz that the NCR Missionaries give can be bugged. For instance answering Vault 13 to the second question will register as a correct, but cause Two-Headed Bear to register as incorrect for the third question. Just make sure to use the three answers listed above. (Note: The first two questions will always have the correct response but if either is wrong the third question will be wrong.) [platforms tag needed]
Question 3 of the NCR loyalty test can be bugged causing all given answers to be wrong, even when selecting the correct one. This problem can be resolved by pickpocketing the note from either missionary. [platforms tag needed]