Question: I find when playing in a minor key I always end up playing the major version of the V chord. (So in Em I always end up playing a "B major" chord.)

However, the third of this chord should be played as a minor when looking at the key and it seems like using an accidental as frequently as I do is not really in the spirit of accidentals.

Am I making a mistake in my music theory? Am I playing the key I think I am? Is this just a mode? Do I need to start playing more B minors?

Please bear in mind I am relatively new to music theory and would appreciate it if you treat me like an idiot. Thanks.

Answer: 
Two points.

First: the scale of a key or mode is a framework. It doesn't instruct you NOT to use other notes. A piece of music that used ONLY diatonic (in the scale) notes and chords would be very basic and possibly very boring (although there's also beauty in simplicity).

Second: in traditional harmony the dominant chord of both a major or minor key is normally major. If we're writing the sort of music that deals in dominant > tonic relationships (and that's the bulk of Western music for the last few hundred years) the dominant chord NEEDS to tend toward the tonic chord. A large part of that tension comes from it including the 'leading note', the note a semitone below the tonic.

We don't HAVE to use major dominant chords. But we don't have to reject then either. You're doing nothing 'wrong' by sharpening that 7th into a leading note. Equally, you're doing nothing 'wrong' by investigating the sound of a minor dominant chord.

"Theory describes, it does not command".
Although the D in the scale of B will be natural in the key signature, it will often be sharpened when leading to the key note E. Think of the melodic minor scale for E. D will be sharpened on the way up but natural on the way down. 

So, if your chord is to resolve to the tonic E, the sharpened D will work nicely. You might find that a B minor chord works well if it goes somewhere else next.