This paper proposes a smartphone-based method for measuring Blood Pressure (BP) using the oscillometric method. For oscillometry, it is necessary to measure (1) the pressure applied to the artery and (2) the local blood volume change. This is accomplished by performing an oscillometric measurement at the finger's digital artery, whereby a user presses down on the phone's camera with steadily increasing force. The camera is used to capture the blood volume change using photoplethysmography. We devised a novel method for measuring the force applied of the finger without the use of specialized smartphone hardware with a technique called Vibrometric Force Estimation (VFE). The fundamental concept of VFE relies on a phenomenon where a vibrating object is dampened when an external force is applied on to it. This phenomenon can be recreated using the phone's own vibration motor and measured using the phone's Inertial Measurement Unit (IMU). A cross device reliability study with three smartphones of different manufacturers, shape, and prices results in similar force estimation performance across all smartphone models. In an N = 24 proof of concept study of the BP measurement, the smartphone technique achieves a mean absolute error of 9.21 mmHg and 7.77 mmHg of systolic and diastolic BP, respectively, compared to an FDA approved BP cuff. The vision for this technology is not necessarily to replace existing BP monitoring solutions, but rather to introduce a downloadable smartphone software application that could serve as a low-barrier hypertension screening measurement fit for widespread adoption.