Is it possible, when looking for a chaplain position, to be able learn everyone's beliefs, and appeal to them as appropriately as possible?

Chaplains are not required to learn or practice multiple denominations. Most will study and practice the denomination they currently follow. During religious studies in college they do become at least familiar with many other religions. What they do all have in common is their training and abilities to counsel those in need.

Your job won't be to know everyone's beliefs. You job will be to provide services and facilities so that they may practice those beliefs. There is a big difference.
When people have specific needs, they will tell you. Have an open mind, Google at you fingertips, and be wiley about finding resources and support in the surrounding community, and you'll do fine.

You are not required to know all of the religions as a veteran and a Chaplains Assistant for 13yrs. That was our job to be familiar with all religions. This back in 1982 when we had MOS testing which we hands on and written test we had to pass for our MOS which as a chaplains Assitant we had to set up for protest, catholic and Jewish services. This was for us to know and the chaplains only needed to practice in their own faith