Keep asking! It never hurts to be tenacious about something you really want. It can be frustrating when you keep hearing a "no" answer, but if you don't keep putting yourself out there and asking, then you definitely won't be getting any shadowing opportunities.
My advice is to call early in the day ( many vet clinics are busy with surgeries or take lunch breaks in the middle of the day). Ask to speak with the veterinarian first. And do your research. If you want to shadow at a certain vet hospital, know the name of the vet you want to shadow. It looks unprofessional, if you just ask to speak with "a veterinarian" rather than Dr. John Doe or whatever his/her name is.
Tell the vet you are pre-vet and what you want to get out of job shadowing. That veterinarian was once where you are now. Many will be sympathetic to your cause of gaining veterinary experience and will listen to you. Be professional, be friendly, and most importantly be yourself. If the vet is unavailable leave a message with the receptionist, and always follow up with another phone call later in the week if you haven't heard anything back.
Keep trying. Many vet clinics have restrictions on job shadowing because of insurance and possible injuries, so try not to take it personally. If you continue to work at it and keep putting yourself out there, you will find a veterinarian that will let you shadow them.