Hello Persefoni,
This is unfortunately a limitation of the labeling engine in ArcGIS Online. ArcGIS Pro has access to the maplex labeling engine, which is capable of more placement analysis & generating labels with more complicated feature clustering.
Please see the following quote from our documentation on labeling considerations in ArcGIS Online:
"Map Viewer places as many labels on the map as possible without overlapping them. Thus, in areas where features are tightly clustered, some features may not get labeled."
- Create labels—ArcGIS Online Help | Documentation
Labeling closely-clustered points will unfortunately not be possible within a webmap due to this limitation. There isn't exactly a data-based workaround for this either, as even if you were to break up the points into multiple layers, there is a chance the webmap viewer would only still label the 'top' layer or point if they are as closely clustered as you describe.
As a workaround, I would suggest creating a 'Map Notes' layer in your webmap. This will allow you to manually add text boxes to create additional labels, which work similar to annotations in ArcGIS Pro. Check out the below documentation for further information on working with Map Notes!
Work with Map Notes:
Map notes—Portal for ArcGIS | ArcGIS Enterprise
(Note: This documentation is for Portal, however the knowledge still applies the same to the AGOL environment.)
I hope this information helps!
-TB
