My knowledge of Star Trek is limited to say the least. I have seen all of the movies, few of the TV show episodes and read none of the books. As such, I am concerned that I may accidentally use the name of an established character. There are only so many combination of letters you can use to make-up a name. SInce I doubt there is an all-encompassing database of every proper noun from all canon and soft cannon Star Trek properties, what am I to do?

There are some powerful databases out there like memory alpha, but I wouldn't worry about it. If you mistakenly call someone Harry Mudd, but the character is a Bolian....

What's the chance that you'll decide to make a blind African American chief of engineering and name him Geordi LaForge?

I agree. I wouldn't worry about it. Check these two links if you are unsure. Just put the name you want to use in the search box and see if anything comes up.