Does radiation therapy weaken people's immune systems?

Yes, but wait.. There is truth that radiotherapy alters immmunity. In the olden days, reactivation of quiescent tuberculosis occured. We note that after radiotherapy, the lymphocyte white cell population disappears or at least is markedly diminished for some time. Surprisingly, it does not lead to definable illness, and the lymphocyte poulation returns. Explains transient psoriasis benefits/changes.
Very little. Radiation acts locally. Depending on the region of the body that is being irradiated it can have some affects on the immune system. Usually this affect is minimal. We rarely need to monitor blood counts if only radiation is being administered. Chemotherapy on the other hand fracture immune system greatly since it is a systemic therapy (not local) and often need to monitor blood counts.