Qubits are the standard basis for quantum computation with many competing host platforms such as superconducting circuits, trapped ions, and quantum dots, to name a few. Part of the recent efforts with these platforms focused on simulations of fermionic systems. However, the mapping from qubits to local fermionic modes is inefficient because it introduces additional overhead to the calculations. To overcome this limitation, we propose a practical implementation of a universal quantum computer that uses local fermionic modes rather than qubits. Our design consists of quantum dots tunnel coupled by a hybrid superconducting island together with a tunable capacitive coupling between the dots. We show that coherent control of Cooper pair splitting, elastic cotunneling, and Coulomb interactions allows us to implement the universal set of quantum gates. Finally, we discuss possible limitations of the device and list necessary experimental efforts to overcome them. Particularly, we predict short coherence times due to charge noise and develop an alternative operational regime using neutral Andreev fermions.