A vaccine for one disease does not make you more susceptible to other diseases.
One common misconception about vaccines is that they weaken a person's natural immunity. A 2018 study found that there was no statistically significant difference in the level of immunity against non–vaccine-targeted infections in vaccinated children compared to non-vaccinated children. This information follows an older study, which evaluated Danish children born from 1990-2001, and found that receiving vaccines did not increase the risk of hospitalization for non-targeted diseases.
Furthermore, a study in Germany of 496 vaccinated and unvaccinated children found that children who received immunizations against diphtheria, pertussis, tetanus and polio within the first three months of life had fewer infections with vaccine-related and -unrelated pathogens than the non-vaccinated group.