The Near Me widget export data action does not always work. Michelle at tech support suggested I provide feedback here as the widget is in beta and tech support does not support beta releases. When you click on the export 'arrow', no corresponding export options (ie to csv etc) appear.
I am having the same problem. Are you using hosted feature layers? For my application I can't use hosted feature layers - but I tested the tool with data uploaded directly to AGOL and the export worked.
I hope they fix this to work for non-hosted layers. Other tools in experience builder export my data without any issues.
I was using a map service layer published to an ArcGIS REST service. I tried a different published map service layer and it DID actually export, so it didn't seem to be entirely a problem with the layer type. I ended up using the query feature to generate an export instead and it worked OK.