Unless people receive a live vaccine, any sickness after injection is due to poisoning, and not because of a low-level infection

After all, the ingredients in non-live vaccines do not contain living viruses or bacteria (at least in theory) and so they shouldn't be able to cause any kind of infection in the body. 

However, as we all have bacteria and viruses in our bodies it means that any poisoning, including by vaccines, can create conditions that lead to such viruses or bacteria becoming active, taking advantage of the bodies weakened state.  This can lead to infections and viral disease occurring as a result of the poisonous shock to the body, but not caused by the pieces of virus or bacteria (or the poisons they create) that are in the "non-live" vaccine.

There are live vaccines with either live viruses or bacteria in them and these can cause infectious illness the person receiving them, if they have a poorly functioning immune system. An example includes the MMR (mumps, measles and rubella) vaccine and certain others.