As far as I know, not through an Esri supported service. However, you can try using the open-source unsupported "ArcGIS Online Assistant" https://ago-assistant.esri.com/ . I've had decent success with it, though it does have its limitations.
There's an idea here that you can 'up-vote'. https://community.esri.com/ideas/5847 This would certainly be an appreciated feature for ArcGIS Online!
Thanks so much for your recommendations. I tried the export to GeoJSON from the feature service export function; it exported to the other organization without the attachment field.
Then I used the Admin Tools for ArcGIS and had no luck. It delivered an error that unfortunately was just a little "Greek" to me.
I then tried your recommendation of the "ago-assistant". It worked well, provided the attachment field with the capabilities of adding photos. The bad is that I already had several images attached to the features and they failed to copy over.
How much content do you need to move from one organization to another? If it is just a few maps, it may be beneficial to download and republish the data from your source organization to your destination organization.
you could then use ago assistant to copy the web map. You would still need to update the service urls to point to the newly published layers.
I have bunches of content (feature layers) with attachments that will need to be moved to the other organization at some point in time. Exporting to FGDB and then republishing seems to do the trick, all images/attachments including the domain drop-down menus publish correctly, so far anyhow. For what I need, this is probably the best workflow until something is added to AGOL that will allow the full feature layer to be moved.
Thanks for your input!!