We propose TRAMBA, a hybrid transformer and Mamba architecture for acoustic and bone conduction speech enhancement, suitable for mobile and wearable platforms. Bone conduction speech enhancement has been impractical to adopt in mobile and wearable platforms for several reasons: (i) data collection is labor-intensive, resulting in scarcity; (ii) there exists a performance gap between state of-art models with memory footprints of hundreds of MBs and methods better suited for resource-constrained systems. To adapt TRAMBA to vibration-based sensing modalities, we pre-train TRAMBA with audio speech datasets that are widely available. Then, users fine-tune with a small amount of bone conduction data. TRAMBA outperforms state-of-art GANs by up to 7.3% in PESQ and 1.8% in STOI, with an order of magnitude smaller memory footprint and an inference speed up of up to 465 times. We integrate TRAMBA into real systems and show that TRAMBA (i) improves battery life of wearables by up to 160% by requiring less data sampling and transmission; (ii) generates higher quality voice in noisy environments than over-the-air speech; (iii) requires a memory footprint of less than 20.0 MB.