Hi Ben,
Are you able to share the service or web map? Is the service hosted in ArcGIS Online or a local ArcGIS Server?
As a first step, I suggest monitoring the request to the attachment using developer tools. This is outlined in this blog: https://community.esri.com/groups/cats/blog/2017/05/04/monitoring-web-service-requests-from-web-apps....
Try finding the request to the specific attachment and launching the url in multiple browsers. If you are able to retrieve the attachment from a direct rest call in all browsers, then then there is no error the service and it may be specific to the browser. This should still be looked into if the attachments only load in some browsers. Are the attachments small or large?
Tech support can also help look into this if your data is private or for specific troubleshooting steps.
-Kelly