Can hand sanitizers weaken my immune system?

Absolutely not. Your immune system is mainly made of antibodies which develop over time. These antibodies hang out in your blood and in lymph nodes, the liver, and the spleen. Wiping something on your hand will not affect these antibodies. If someone tells you “alcohol-based hand sanitizers weaken your immune system” that would be like them saying “hey be careful with alcohol-based hand sanitizers, I hear they can get you drunk if you rub it on your skin” Of course that is not true. The hand sanitizer barely penetrates your skin. Ditto for why it does not affect your immune system.