The paper is devoted to an experimental investigation of distributed receptivity of a laminar swept-wing boundary layer to unsteady freestream vortices with streamwise orientation of the vorticity vector. The experiments were performed on a model of a swept wing with sweep angle of $25^{\circ }$ at fully controlled disturbance conditions with freestream vortices generated by a special disturbance source. It is found that the unsteady streamwise vortices are able to provide very efficient excitation of non-stationary cross-flow instability modes without the necessity of the presence of any surface non-uniformities. The developed experimental approach provides the possibility for a detailed quantitative investigation of the mechanism of distributed excitation of unsteady boundary-layer disturbances due to scattering of freestream vortices on natural base-flow non-uniformity. This mechanism has been studied experimentally in detail. This paper (Part 1 of the present study) is devoted to description of: (a) the experimental approach and the base-flow structure; (b) the method of excitation of fully controlled streamwise-elongated freestream vortices; (c) the results of measurements of structure of these vortices; and (d) the experimental evidence of high efficiency of the distributed vortex receptivity mechanism under study. Part 2 of this study (see Borodulin et al., J. Fluid Mech., vol. 908, 2021, A15) is devoted to the theoretical background and experimental quantitative characteristics of the distributed vortex receptivity. Values of the corresponding receptivity coefficients are estimated there for their three different definitions as functions of the disturbance frequency, spanwise wavenumber and wave propagation angle.