This tune is in A mixolydian. This means that the complete list of chords with which you would expect to accompany it are the same ones available in D major, but beginning and ending from the A chord. Therefore your complete list of likely chord choices would be A major (or A7, as this would be chord V in D ionian), B minor, C# diminished, D major, E minor, F# minor and G major.
The melody uses a trick common in Irish tunes and changes the flattened seventh note from the A mixolydian scale to the normal seventh from the scale of A major / ionian in the final bars of each section. This means that your chord V becomes E major in these bars only.
Bar 3 has to take a G chord (chord VII) or its related minor, Em, because the tune notes clearly outline a G chord (G B D G – -). Bar 4 more or less outlines a B minor chord or its related major, D (B D G B – -). I have plumped for D as I prefer its bright, optimistic sound and it is easier to play!
Bars 1 and 2 of the B part contain the same notes, and hence the same chords, as bars 3 and 4 of the A part.
The tune is somewhat unusual in that the fourth bar of the B part does not have a chord V section, but simply outlines an A chord throughout. However the section does finish with a clear chord V – I cadence as usual. You can tell that the final bar of the A part needs chord five as it contains the seventh note of the A major scale. The seventh note is a very unstable tone and appears in the fifth chord- A major’s seventh tone is G#, which is the major 3rd of A’s chord V, E major. For this reason you can usually take it as a given that a bar containing this note on a dominant beat or for any longer than a quaver is a chord V bar.
In this version I have inverted the D chord in the fourth and eighth bars so that its 3rd, F#, is at the lowest pitch within it. This makes my chords sound more coherent as the large jump in bass notes from G all the way to D is replaced by a smaller gap of 1 semitone from G down to F#. To play this chord, make a D major shape and then loop your thumb around the back of the neck in order to fret the second fret of the bottom E string.
In this version I have substituted A to A7. This works because the mode of A mixolydian contains the same notes, and therefore chords, as the mode of D ionian (major). In D major, A would be chord V and when we begin to use tetrads, four note jazzy chords with 7ths in them, the V chord in a major key is always a dominant chord. This also means that the I chord in the mixolydian mode is always dominant.
Chord VII, G, has been substituted to G7. This is because in every mode other than ionian, chord VII becomes a dominant 7 chord if tetrads are being used. Strictly speaking, the D (IV) chord in the fourth bar should be a major 7 type chord, but I have borrowed the use of a dominant IV chord from blues. This works ok, because the note in the D9 chord which is not in the scale of A mixolydian is a C♮, and there is no C# in the tune in this bar.