A tunable magnetic field at low temperatures is essential for numerous applications, including spintronics, magnetic resonance imaging, and condensed matter physics. While commercial superconducting vector magnets are available, they are complex, expensive, and often not adaptable to specific experimental needs. As a result, simple in-house designs are often being used in research environments. However, no comprehensive step-by-step guide for their construction currently exists. In this work, we provide a detailed manual for designing and building a cryogenically compatible three-axis vector magnet. The system is tested at the mixing chamber of a dilution refrigerator at temperatures ranging from 15 mK to 4 K, with no significant increase in base temperature. Safety measures are implemented to mitigate heating from quenching. The coils are successfully driven with DC currents as high as 3 A, generating magnetic fields of up to 2.5 T in the bobbin’s bore and 0.4 T at the sample position. Magnetic field measurements using Hall sensors demonstrate good agreement with the predictions of the designed performance.