I cannot speak to Mizzou's opportunities for Shelter Medicine, but I have been overwhelmingly impressed with LSU's Shelter Medicine program! There are so many ways to get involved from day one and to keep learning throughout the entire four years.
The first two things I recommend are joining the Shelter Medicine Club and coming to Spay Day. The Shelter Medicine Club doesn't have wet labs in the traditional sense. Instead, there are shelter workdays, and that varies with respect to the shelter and what they need. The club has two to three lunch meetings per semester with speakers on various topics.
Spay Days are monthly, and it is a partnership with a local rescue to care for 1) trap, neuter, release 2) shelter cats and 3) cats whose owners could not otherwise get veterinary care for their cats. It's great experience, and you are helping provide a great service to the community.
Then, in spring of your first year you can take the Shelter Medicine elective. I took it my first year and loved it. We got to tour an innovative local shelter, and we had extremely interesting speakers.
When you get into clinics, there is a Shelter Medicine rotation, there are externship options, and there is an external rotation at a local shelter (it's classified as an elective).
If you have any more specific questions, feel free to ask or to direct message me.