Hair care robots have the potential to alleviate labor shortages in elderly care and enable those with limited mobility to express their identities through hair styling. We present MOE-Hair, a system that incorporates a compliant soft robotic manipulator and visual method for applied force estimation to perform three hair-care skills: head patting, finger combing, and hair grasping. MOE-Hair underscores two advantages of soft robotic manipulators in hair-care applications: safety through mechanical compliance and sensing force through observing deformation. We introduce a tendon-driven soft robotic end-effector called Multi-finger Omnidirectional End-effector (MOE) with a wrist-mounted RGBD camera for hair-care applications. We also introduce a method to infer MOE's applied forces on the contacting surface from observed deformations and tendon tensions from the actuators. We tested the system on a force-sensorized mannequin head to determine that the system was safe for human users and evaluated the components of the system. Then, we performed a user study with 12 participants to evaluate user perception. We found that in hair-grasping tasks, MOE exerts less force on a force-sensorized mannequin head while being able to grasp a similar amount of hair compared to rigid grippers. We demonstrate on the mannequin head that we can infer about the applied forces in hair-care settings with a reduction in error up to 60.1\% and 20.3\% respectively for actuator current load-only and depth image-only baselines by incorporating both as observation to the model. We found statistically significant results in the user study to evaluate the system on comfort, effectiveness, and perceived appropriate use of force, showing that the participants preferred the proposed system. The results suggest that soft robots are uniquely advantaged in contact-rich hair-care tasks and that reasoning about the fingers' applied forces is important despite MOE's inherent compliance.