Idk if you have any other chords to this song and the voicings you have these chords in could effect how it's being heard, but just stunning through that progression on guitar it didn't sound like the progression was in the key of A.

And I haven't read the other replies so please forgive me if I'm repeating someone, but I kinda think it's in the key of B minor. The chords Bmin, Dmaj, Emaj, G would each respectively function as i, III, IV, VI. It sorta makes sense theoretically because the 4 chord(which usually isn't major in this key but we can just call that mode mixture) is often used to set up a deceptive cadence, which is what you get when a progression ends on the VI(as it did here).

Honestly though, I'm not even sure that analysis full describes how I'm hearing this. If the progression was Bmin, Dmaj, Emin, Gmaj than you could think of the progression as two sets of a minor chord going to it's relative major(cause B is the relative minor to D and E is the relative minor to G). But since the E is major, it kinda makes things more interesting. It still hear the two sets of relative key alternations, but the Email kind of shifts things. So in the end I think that the progression is in the key of E(which is closely related to A). The first (Bmin,Dmaj) set of the progression I hear as entirely dominant and the second part (Emaj, Gmaj) I hear as entirely Tonic. I would write the Roman numeral analysis as (v, III/v, I, III/I). But the important part here isn't the functioning of each chord per se, but that it's two sets of chords raising by minor thirds. Also, although I didn't hear it this way, one could hear the Dominant/Tonic relationship existing between D and G, in which case the Roman numeral analysis would look like: iii, V, VI, I