At the turn of the twentieth century, children received one vaccine: the smallpox inoculation. Now they routinely receive 11 different vaccinations, in 20 different injections, before their second birthday. This is bound to have an effect.
When a disease or virus enters the body naturally, through inhalation, for example, the vaccine takes much longer to attack the nervous system. When it is injected, the nervous system is instantly exposed to a large dose of the disease. A child’s underdeveloped immune system cannot produce the necessary immune response to a vaccine. As a result, they are more likely to develop infections and other diseases while their immune system is in a weakened state. 
It is not the number of vaccines that overwhelms the immune system, it is the number of immune-alerting antigens. In this area, modern vaccines are exceptionally safe. The first smallpox vaccine developed by Edward Jenner had 200 different antigens that stimulated the immune system. Today, all 15 vaccine doses children receive contain a combined total of 150. Extensive studies have been carried out on the possibility of vaccines overloading a child’s immune system. None have shown that the quantity of immune-alerting antigens causes a problem. Children develop the ability to produce humoral and cellular responses to most vaccinations between two and six months of age. By six months of age, the child’s immune system can theoretically sufficiently respond to 10,000 at one time. In other words, 0.1% of a child’s immune system is “focused on” fighting the vaccination, leaving the remaining 99.9% to fend off other pathogens and infections (even this is a conservative estimate as B- and T- cells are constantly being replenished, meaning the vaccine will never “use up” 0.1% of the immune system’s capabilities). 
[P1] Vaccines overwhelm a child's immune system. [P2] Therefore, the child is vulnerable to other diseases and infections. [P3] Therefore, vaccines are dangerous.
[Rejecting P1] A child's immune system is capable of producing the B-and T- cells necessary to handle vaccinations at six months of age. [Rejecting P2] Therefore, the child is not vulnerable to disease. [Rejecting P3] Therefore, vaccinations are not dangerous.