Seismic urgent computing enables early assessment of an earthquake's impact by delivering rapid simulation-based ground-shaking forecasts. This information can be used by local authorities and disaster risk managers to inform decisions for rescue and mitigation activities in the affected areas. Uncertainty quantification for urgent computing applications stands as one of the most challenging tasks. Present-day practice accounts for the uncertainty stemming from Ground Motion Models (GMMs), but neglects the uncertainty originating from the source model, which, in the first minutes after an earthquake, is only known approximately. In principle, earthquake source uncertainty can be propagated to ground motion predictions with physics-based simulations of an ensemble of earthquake scenarios capturing source variability. However, full ensemble simulation is unfeasible under emergency conditions with strict time constraints. Here we present ProbShakemap, a Python toolbox that generates multi-scenario ensembles and delivers ensemble-based forecasts for urgent source uncertainty quantification. The toolbox implements GMMs to efficiently propagate source uncertainty from the ensemble of scenarios to ground motion predictions at a set of Points of Interest (POIs), while also accounting for model uncertainty (by accommodating multiple GMMs, if available) along with their intrinsic uncertainty. ProbShakemap incorporates functionalities from two open-source toolboxes routinely implemented in seismic hazard and risk analyses: the USGS ShakeMap software and the OpenQuake-engine. ShakeMap modules are implemented to automatically select the set and weights of GMMs available for the region struck by the earthquake, whereas the OpenQuake-engine libraries are used to compute ground shaking over a set of points by randomly sampling the available GMMs. ProbShakemap provides the user with a set of tools to explore, at each POI, the predictive distribution of ground-motion values encompassing source uncertainty, model uncertainty and the inherent GMMs variability. Our proposed method is quantitatively tested against the 30 October 2016 Mw 6.5 Norcia, and the 6 February 2023 Mw 7.8 Pazarcik earthquakes. We also illustrate the differences between ProbShakemap and ShakeMap output.