In this paper, we consider the problem of designing a distributed controller for a formation of spacecraft following a periodic orbit. Each satellite is controlled locally on the basis of information from only a subset of the others (the nearest ones). We describe the dynamics of each spacecraft by means of a linear time-periodic (LTP) approximation, and we cast the satellite formation into a state-space formulation that facilitates control synthesis. Our technique exploits a novel modal decomposition of the state-space model and uses linear matrix inequalities (LMIs) for suboptimal control design of distributed controllers with guaranteed performance for formations of any size. The application of the method is shown in two case studies. The first example is inspired by a mission in a low, sun-synchronous Earth orbit, namely the new Dutch-Chinese Formation for Atmospheric Science and Technology demonstration mission (FAST), which is now in the preliminary design phase. The second example deals with a formation of spacecraft in a halo orbit.