Just a word of clarification. It's important to distinguish between pre-exposure and post-exposure vaccines
Pre-exposure vaccines (Poliomyelitis MMR, Varicella etc.) do not come from animals. The viral agent which forms the basis for the vaccine is cultured in egg. The most effective pre-exposure vaccines contain a live attenuated (weak) strain of the virus. When an attenuated strain is not available, the virus is "killed," ususally so that only the protein shell remains.
Either way, your body will react basically the same as if it were attacked by the real thing. Over the next few weeks it will produce anti-bodies against the virus, giving you active immunity.
Post-exposure vaccines are altogether different and not really the topic of this thread. Lets say you step on rusty nail and its been more than five years since your last tetanus booster. Let's say you are bitten by a rabid animal. You need immediate passive immunity.
Post-eposure vaccines contain immune-globuliln --that is to say antibodies already made against the virus. Many common immune-globulins are produced recombinately today, but in the old days, the only source was some other creature that had already been exposed. It was common to use horses. Jehovah's Witnesses used to have a big problem with post-exposure vaccines.
Regardless of the source, immune-globulin is a plasma protein. It is not cellular and it doesn't carry disease.