This tune is in E dorian, which contains the same chords as D ionian. This means that the chords available are D major, E minor, F# minor, G major, A minor, B minor and C# diminished. The second bar contains an outline of a D major chord for its second half (A F# D, aka the notes of a D major triad in reverse order) so I have put a D chord there. Chord V (Bm) would normally go on every seventh foot tap and resolve back to chord I (Em) at the ends of sections, but I have replaced this with its related major, D, because this is easier to play. This can be done in any mode which has a minor chord V, aka any mode other than ionian.
In the B part, bar 2 contains the notes F# A G F# E D. As the dominant beats (marked in bold) are both F# notes, which appear in chord IV, D (D, F# and A) I have put chord IV in this section. Rather than have another block of Em straight after, I have substituted the Em to its related major, G major. Bars 7 and 8 of the B part contain the notes “E D C# D – A B A G F# G A”. This is a decidedly chord V sounding section, with D, B, F# (notes from D or Bm chords) making up its dominant beats. Hence I have put two bars of D here instead of sticking to the usual formula of just putting chord V (Bm) or its related major (D) in the 8th bar. In fact playing Em for bar 7 and Bm or D for bar 8 would have worked equally well, but I like the sound of the extra bar of D… And they’re my chords and I can do what I want, so there.
In this version I have switched the chords in bar 4 to be G and A. This provides a bright lift from the E minor root chord and works well as the notes are “D C# B” for the first half (a G chord is G, B and D) and “A F# D” for the second half (the dominant A note sounds cool with an A chord). I have substituted the E minor in bar 7 to its related major, G. This works well transitioning into D (chord V) as any progression which resolves down a fifth will always work well. If you wanted a jazzy flavour you could try replacing the G with a G7, as any time a chord resolves down a fifth you can replace it with a 7 chord for jazzy effect.
I have ended the B part on an A chord. This is a pretty nifty little trick which I like to use in the dorian mode- if you end a section on chord IV it gives the tune a far away, make-believe quality, as if it very nearly achieved its hopes and dreams of dorian corporate supremacy, but then right at the point of success, suddenly decided to give up on the whole thing and go off to the other side of the planet, climb a mountain and become a yak farmer instead. If you want another interesting version of this trick, try beginning a section of a dorian tune on its chord IV and then alternating between this and chord V.
In this version I have used tetrads (7 chords) instead of triads (normal ones). The tetrads available are the same as the ones which would be available in the key of D ionian, aka D major 7, E minor 7, F# minor 7, G major 7, A dominant 7, B minor 7 and C# ½ diminished (in place of which we would usually play D/F#, aka D with a thumb). Additionally, in any mode other than the ionian, we make the VII chord into a dominant 7 too. That means that our complete list of available tetrads for E dorian should be: E minor 7, F# minor 7, G major 7, A dominant 7, B minor 7, C# ½ diminished and D dominant 7. In bar 7 of the A part I have borrowed a C major 7 chord from the aeolian mode. This works because the sixth note of the E dorian scale is not present in this bar, so I can use chords from either the E dorian or the E aeolian mode as only the 6th note differs between the two.
In the B part I have borrowed an A minor 7 chord from E aeolian and used it to create tension by deliberately avoiding chord I. You can always substitute chord IV in a chord I section, but in this case I have deliberately not played chord I at all, in order to make the section feel like it has moved away from its key chord. There is no 6th note in the B part of the tune until the penultimate bar, at which point I am playing chord V anyway which is the same in both aeolian and dorian, so I have treated the whole section as though it were written in the aeolian mode.