bbPress.org

Doesn’t this require the person to put the image somewhere – ie on Smugmug – and then use the IMG tags to reference that image?

Yes

I was looking for something that allows you to just decide to show a photo in a forum post and upload it then and there to the site and have it display inside the post.

For this a common a plugin is this one https://wordpress.org/plugins/gd-bbpress-attachments/