Infants are born with an immature innate immune system. Since their immune systems are weaker than in adults, they are less likely to develop an anaphylactic reaction. A possible mechanism for the formation of anaphylaxis in newborns has not yet been elucidated.
This paper describes the first case of anaphylactic shock due to vitamin K injection.
Additional sources: 
Safety of the MMR vaccine was discussed in detail in measles and mumps sections. Here are some more studies, focused more on rubella:
The risk of anaphylactic shock due to vaccination is 1.89 in 10,000 cases for measles vaccine and 2.24 in 10,000 cases for rubella vaccine. The authors believe that these figures are underestimated, since the exact number of administered vaccines is unknown, and that the actual figures may be 3-5 times higher.
The risk of anaphylactic shock due to MMR vaccine was estimated as 1.4 in 100,000 cases in 2004. However, in 2003 the risk of anaphylactic shock due to all the vaccines was estimated at 0.65 in a million.
The concept of allergies was unknown until 1906. It was invented by an Austrian pediatrician and used to describe the strange symptoms he observed in those who received the diphtheria immunoglobulin.
The concept of anaphylactic shock also did not exist until 1902.
An IOM report, in which, among other things, is stated that they have found a causal relationship between tetanus/diphtheria vaccine and Guillain-Barré syndrome, anaphylactic shock and brachial neuritis.Adverse Reactions Following Immunization with MMR Vaccine in Children at Selected Provinces of Iran. 2011, Esteghamati, Arch Iran Med
A study of the side effects of MMR in Iran (43,000 vaccinated children). 1.8% of the vaccinated contracted mumps as a result of the vaccination. Two suffered encephalopathy and two more had anaphylactic shock. That is, the risk of encephalopathy and anaphylactic shock is about 1 in 20,000, rather than "one in a million", as usually claimed.
The risk of convulsions (not febrile) was 1 in 2,000 vaccinations in infants. The risk of febrile seizures was 1 in 1750.