The American Academy of Pediatrics, based on a review of current published studies, finds no no support for the idea that multiple vaccines “overwhelm, weaken, or ‘use up’” the immune system. “On the contrary, young infants have an enormous capacity to respond to multiple vaccines.” A more recent study, published in 2018 in the American Medical Association's journal, supports the finding that multiple vaccinations don’t weaken a child’s immune system.
The Centers for Disease Control also notes that, in many cases, infants are administered combinations of several recommended vaccines in a single shot. These combination vaccines have been found to be an efficient way to get infants and other children vaccinated.