Sepsis is a significant cause of early mortality, high healthcare costs, and disability-adjusted life years. Digital interventions like continuous cardiac monitoring can help detect early warning signs and facilitate effective interventions. This paper introduces i-CardiAx, a wearable sensor utilizing low-power high-sensitivity accelerometers to measure vital signs crucial for cardiovascular health: heart rate (HR), blood pressure (BP), and respiratory rate (RR). Data collected from 10 healthy subjects using the i-CardiAx chest patch were used to develop and evaluate lightweight vital sign measurement algorithms. The algorithms demonstrated high performance: RR (-0.11 $\pm$ 0.77 breaths\min), HR (0.82 $\pm$ 2.85 beats\min), and systolic BP (-0.08 $\pm$ 6.245 mmHg). These algorithms are embedded in an ARM Cortex-M33 processor with Bluetooth Low Energy (BLE) support, achieving inference times of 4.2 ms for HR and RR, and 8.5 ms for BP. Additionally, a multi-channel quantized Temporal Convolutional Neural (TCN) Network, trained on the open-source HiRID dataset, was developed to detect sepsis onset using digitally acquired vital signs from i-CardiAx. The quantized TCN, deployed on i-CardiAx, predicted sepsis with a median time of 8.2 hours and an energy per inference of 1.29 mJ. The i-CardiAx wearable boasts a sleep power of 0.152 mW and an average power consumption of 0.77 mW, enabling a 100 mAh battery to last approximately two weeks (432 hours) with continuous monitoring of HR, BP, and RR at 30 measurements per hour and running inference every 30 minutes. In conclusion, i-CardiAx offers an energy-efficient, high-sensitivity method for long-term cardiovascular monitoring, providing predictive alerts for sepsis and other life-threatening events.