Step 1. Find a vet
Step 2. Dress decently before going in
Step 3. Bring resume/CV/cover letter
Step 4. Ask to shadow in person, if you can't talk with the vet, talk with the receptionist, explain to him/her what you are looking for, leave your resume/CV and cover letter (this should explain what you are looking to do). Do NOT ask to shadow for a summer, a month, or even a week, ask for a single day. Seriously. Do not get angry if they answer is no, say "thank you for your time" and move along to the next clinic.
Step 5. Shadow, be present, ask questions, but don't ask questions at the wrong time (such as in front of clients, while the vet is in the middle of surgery/CPR/on the phone), find a time when the vet seems to be not as busy and say "I have some questions for you would you have time if I ask you those now or could you let me know when you would be available?"
Step 6. If you are shadowing, as hard as it is, do NOT do NOT do NOT attempt to "jump in and assist", stand back, watch, ask questions and then ASK if there is anything you can help with. You may not be legally allowed to do anything... including picking up dog ****. Serious.
Step 7. If the one day goes well, tell the vet, "thank you so much for this opportunity, can I ask you about a possible more long-term shadowing position?"
Step 8. Return as needed/if they say yes.