Yes, any patch she receives goes on the back of her vest or sash.

Girl scout day is this sunday.

https://www.mlb.com/nationals/tickets/specials/youth-days

I know a Scout in Scouting America would not wear a Girl Scout patch or pin on their uniform, they are not a part of that organization. I can't imagine that Girl Scouts wants their kids wearing patches and pins from Scouting America. I doubt too many people would make a huge fuss over it though.

If she is going to her brothers events and enjoying them she could join a Cub Scout pack herself and earn the patches and pins for herself.

DD is a girl scout. She will attend national scout day, the one at DC baseball nationals game. I believe this is not a girl scout event, but a cub scout/boy scout event. If she receives a patch/badge or pin from that scout event, can she put it on her girl scout uniform? She asks me if she can wear girl scout uniform that day or it is better for her to wear her brother's scout T shirt (we have extra) that day or regular clothing that day.

I sometimes buy her some fun patches online to put them on her uniform when she attends some meaningful cub scout events.