This study describes the design and implementation of a web-based infrastructure built on open-source components to enable the computation of accessibility scores. It can be adapted to any form of service provision that is represented as geo-located points. The solution aims to extend previous attempts to measure and analyse geographical accessibility in a web-based environment. This design integrates a new approach for generating the variables required in calculating accessibility scores, while allowing the whole system to potentially remain as an open-source and open-data solution. The design allows accessibility scores to be computed using simultaneously both public and private transport modes, thus presenting a more accurate assessment than that obtained from standard single-mode solutions. The infrastructure is described through a consideration of each fundamental component, demonstrating how travel distances and alternative journeys are generated and applied to the accessibility model. Results produced by the solution enable the comparison of accessibility scores generated from alternative modes of transport and are illustrated using a case study to investigate access to sports facilities in Wales. This demonstrates how the system can be used to guide further investment and the future management of service infrastructure.