Quite many schools have shadowing or clinical volunteering or both. Starting with the two schools I know best, as a parent (I am a physician, my college kids are not premeds but many of their friends are and their schools are well known for their premed/prehealth preparation):
(I could not get it to link full Penn med list of supported clinical opportunities for students but if you read this site the links are there– Penn Med and CHOP (peds) are on campus and there are numerous opportunities listed)
Here are a few more as examples: WashU, JohnsHopkins, Harvard, Emory: I stopped searching after finding established shadow/clinical volunteering experiences for undergrads at almost all of the schools I know to have excellent premed prep (ie the twenty or so I would have encouraged my kids to consider if they were premed, because these fit them in other ways, in addition to William and Mary and UVA which are excellent for premed and in-state for us).
When you read further on these websites they all encourage real clinical experience over just observation, and they discuss programs for their students at their own hospitals as well as options in the area.
I echo the other advice that you should focus on where your kid will academically and socially thrive, which you will not know until you see how the high school classwork goes and have scores. Then make a big list and go through each and see what the shadowing/clinical volunteering is like.