There is no provision available out of the box to track the user details. However, you can create a menu macro to run this custom report and add a log entry inside the script using something like this. Here is a sample code from the existing menu macro 'Run Task Board Report' in ALC. Highlighted the code to add log entries.
var itemSelection = VegasClient.getItemSelection(selectedItemsStoreId, selection, selectionImplied, securityContext);
var selItem = itemSelection.get(0);
var names = java.lang.reflect.Array.newInstance(java.lang.String, 1);
names[0] = "projectId";
var vals = java.lang.reflect.Array.newInstance(java.lang.Object, 1);
vals[0] = java.lang.Long.valueOf(selItem.getId());
var types = java.lang.reflect.Array.newInstance(java.lang.String, 1);
types[0] = "integer";
VegasClient.createLogEntry('',' ' ,'Custom Report <Custom report name> is executed','',user);
ClientMethods.openReport(1254, names, vals, types, securityContext);