The disperse markers tool works with representation symbology only. Representations can only be added to feature classes in a geodatabase, so you would need to convert your shapefile to a geodatabase feature class, add it to ArcMap and symbolize it, then convert symbology to representations. Then you could use the disperse markers tool.
Can you use the Disperse Markers tool in ArcGIS Online? I can symbolize the dispersed points in ArcMap, but as soon as I export the data, the representation symbology is lost. Therefore, I cannot upload the dispersed markers into ArcGIS Online where I would like to use the data. Is there another way of representing overlapping data points in ArcGIS Online?