There are no formal shadowing programs that exist at Barnes-Jewish Hospital outside of the MedPrep II course; however, there are some options that do exist in general:
If you do volunteer at a hospital, although you may not shadow specifically, you will get to interact with patients and families throughout their hospital stay. You may even get to introduce yourself to part of the physician team, and you can request a shadow shift with them once you get to know the team.
Research is another good way to make contacts in the hospital, especially if you are engaged in clinical research. Again, once you are a trusted part of the team, approach the physician/PI and ask if you can spend a shift or two with them.
You can contact your own physician (current or former) and ask to spend a day shadowing them. Your pediatrician or family physician knows you and has watched you grow and will likely be thrilled that you are going into medicine like they did and will often be happy to assist.
You can join an online alumni group (Linkedin, Facebook) and look for keywords such as medicine. Introduce yourself and state that you are a current WU student and what your goals are. The Career Center or alumni office may also be of assistance in locating area alumni.
Although you can cold call faculty, it is very difficult to accommodate undergraduate shadowers, and those opportunities are usually reserved for medical students. Please do not send out a mass email to the entire department, as it will likely get you flagged as spam and will lessen your chance of getting a positive response.