I learned to play guitar a couple years ago, but I can't for the life of me play a "B" chord. Is there any version that isn't a bar chord? Or do I have to suck it up and learn to play a bar chord?
Yes, in a nutshell. But no, not a nice open chord like E (022100), if you're in standard tuning.
B7 has its own non-bar opportunities (see Tim's comment). But I'm going to just stick to B major chords with standard tuning, since that is the original question.
Here are four possibilities; some are easier than others:
If you are willing to bar a little, there are more too, like xx9877.
These aren't the only ones out there, but hopefully are a good start. Feel free to comment if you'd like more.