Does my religion (Judaism) allow me to be an organ donor?

The simple answer is yes. But in Judaism there is rarely a simple answer so it is always important to check with your rabbi in the denomination of Judaism that you are part of to discuss the unique context of each case. 

On the whole, there is an overarching principle of pekuach nefesh, the priority of saving a life. Being an organ donor allows you to save a life, and in that case, it is ok. It gets very intricate into the levels of Jewish law about the state that the body needs to be in and what constitutes death and all of that is quite complex, but on the whole, yes. We allow for organ donation. And there are boundaries around how.

Absolutely yes. As organ donation has become a real thing in recent decades, some very significant thinkers and legalists in the Jewish tradition have come up with great reasons why to prioritize organ donation.