Yes, a child's immune system develops and improves over time. However, the idea that the immune system can be "overwhelmed" by vaccinations is specious.
An infant has both passively acquired and adaptive immune responses.
Our immune system helps protect us from pathogens with its multiple layers of varying purpose and specificity. The immune system becomes “educated” as we are gradually exposed to the environment. One branch of the immune system, which is highly adaptive, involves antibodies called “immunoglobulins”. Immunoglobulins are divided into five classes – IgM, IgG, IgA, IgE and IgD – each of which has evolved to deal with different antigens. The capacity of the body to produce each of these varies with age.
Passive immunity refers to antibodies present in breastmilk that provide the infant with temporary immunity to diseases to which the mother has been exposed. This can help protect the infant against infection during the early months of childhood. Maternal IgG gradually disappears during the first 6 to 8 months of life, as the rate of the infant's IgG synthesis increases. Adult concentrations of total IgG are reached by about 7 to 8 years of age.
Adaptive immunity develops throughout life as the result of exposure to pathogens and through immune response to vaccinations. Newborns begin to synthesize IgM antibodies very soon after birth in response to their environment. The premature infant appears to be as capable of doing this as the full-term infant. At about 6 days after birth, the serum concentration of IgM rises sharply and continues to rise until adult levels are reached by approximately 1 year of age.
Exposure to common infectious pathogens thus has an important function in building the child’s healthy immune system via adaptive immunity. A variety of infectious agents, both viral and bacterial, are commonly encountered by children in daycare, resulting in a range of illnesses including upper and lower respiratory tract infections, acute otitis media, gastrointestinal illness, and certain skin ailments. The young child first entering daycare – especially a child with no older siblings – essentially encounters an assault on the immune system that is a direct result of increased environmental exposure. Children under 1 year of age may be particularly hard hit as they are still in the early stages of immune system development.
The AAP article linked at the top of this answer expands on that, indicating that Maternal IgG is transported across the placenta before birth, and maternal secretory IgA is present in breast milk and colostrum. It also offers the qualification that maternal antibodies offer limited immunologic protection when compared with protection afforded by an infant's active immune response.
Another key difference between adult and infant immune responses lies in B-cell responses. Infant B-cell responses are deficient when compared with older children and adults. Until they are about 2 years old, infants have a B-cell response to T-cell-independent antigens (such as polysaccharides) that is considerably less than that found in adults (Rijkers GT, Dollekamp EG, Zegers BJM. The in vitro B-cell response to pneumococcal polysaccharides in adults and neonates. Scand J Immunol. 1987; 25 :447-452).
Infants can, however, generate all functional T-cells, and respond well to antigens (such as proteins) that require T-cell help for development.
Regarding the claims that too many vaccines can overwhelm an immature immune system, that simply does not appear to be true. Infants are capable of protective immune response to vaccines within hours of birth, and are fully capable of generating protective immune responses to multiple vaccines simultaneously.
Approximately 90% of infants develop active protective immune responses to the primary series of diptheria-tetanus-acellular-pertussis, hepatitis B, pneumococcus, Hib, and inactivated polio vaccines given between 2 months and 6 months of age (Plotkin SA, Orenstein WA. Vaccines. 3rd ed. Philadelphia, PA: WB Saunders Co; 1999)
A more practical way to determine the diversity of the immune response would be to estimate the number of vaccines to which a child could respond at one time. If we assume that 1) approximately 10 ng/mL of antibody is likely to be an effective concentration of antibody per epitope (an immunologically distinct region of a protein or polysaccharide),39 2) generation of 10 ng/mL requires approximately 103 B-cells per mL,39 3) a single B-cell clone takes about 1 week to reach the 103 progeny B-cells required to secrete 10 ng/mL of antibody39 (therefore, vaccine-epitope-specific immune responses found about 1 week after immunization can be generated initially from a single B-cell clone per mL), 4) each vaccine contains approximately 100 antigens and 10 epitopes per antigen (ie, 103 epitopes), and 5) approximately 107 B cells are present per mL of circulating blood,39 then each infant would have the theoretical capacity to respond to about 10 000 vaccines at any one time (obtained by dividing 107 B cells per mL by 103 epitopes per vaccine).
However, because naive B- and T-cells are constantly replenished, a vaccine never really “uses up” a fraction of the immune system. For example, studies of T-cell population dynamics in HIV-infected patients indicate that the human T-cell compartment is highly productive. Specifically, the immune system has the ability to replenish about 2 billion CD4+ T lymphocytes each day. Although this replacement activity is most likely much higher than needed for the normal (and as yet unknown) CD4+ T-cell turnover rate, it illustrates the enormous capacity of the immune system to generate lymphocytes as needed.