Orbitronics exploits orbital angular momentum to generate and manipulate angular-momentum currents for controlling magnetisation, offering new approaches to magnetic memory and logic. A quantitative description of this process requires establishing how orbital angular momentum contributes to magnetic moments and, consequently, to the gyromagnetic response that governs magnetisation dynamics. Here, we present a first-principles framework for calculating the effective gyromagnetic tensor (g') of paramagnetic metals, consistently incorporating local and itinerant orbital contributions, spin-orbit coupling, and both state- and occupation-change effects. Applying the framework to V and Pt, we find that orbital paramagnetic responses are comparable to or larger than spin contributions and substantially modify g' from the free-electron value. The atom-centered approximation can underestimate the orbital response by approximately a factor of two, while neglecting occupation changes produces significant errors in the calculated g'. Our calculated g' values for V and Pt agree with experiment, while their pronounced dependence on band filling reveals a route to engineering angular-momentum responses through electronic-structure control. The results establish the modern theory of orbital magnetism as an essential ingredient for describing gyromagnetic response and provide a framework for identifying materials with enhanced orbital functionalities.