Hello @Ivy_Hinson
From what I can understand you want to utilize the "Share a Web Map" option in ArcGIS Pro and share it to ArcGIS Online where the feature class you are working with has "Attachments enabled".
As per my understanding that can be achieved. To give a brief idea, when we share a web map, it will publish the following (In case of ArcGIS Online):
The hosted Feature layer will contain the attachments which you are mentioning. For more detailed understating, I would recommend following documentations:
Hope I was able to assist with this.
-Archit