Does my reputation change this? Whenever I arrive in the morning, I just have the option to "come closer" which usually just leads to a sexual encounter.
The option to sneak in the camp is only after you talk with Jester about talking with Scar. If you made everything (Track event with guard, camp event with tank, an event with Logan - if encouraged to go to the camp then in the camp, if discouraged then in town), there should be a new conversation topic with Jester that will unlock "Base infiltration" option.