Can a 13 year old get a Saturday job in a Veternary Surgery?

Honestly, 13 is too young to work at a vet clinic - you don't have the physical strength to handle animals by yourself and it is a dangerous work environment. There are biological hazards such as feces, urine, infection and blood potentially on every surface. There are many sharp objects, ranging from needles to scalpels, that are in use in many places. The medications being dispensed can cause severe reactions or death, particularly if you accidentally stick yourself with them. Many vet clinics have an X-ray machine, which exposes you to radiation while your body is still growing and developing.

I generally suggest people to start applying for a position at a vet clinic no younger than age 16, and to not work alone until you are 18. There are some exception to this, but I have seen and heard of too many issues with young people getting in over their heads to be comfortable with a 14 year old taking a Labrador out for a walk.