"What keys contain both E and Am"... "By 'keys contain' I mean in the normal rules of choosing what chords 'fit' within a given key"
Strictly speaking, (perhaps surprisingly) the answer is: none. That is, if you only allow chords from the diatonic scales, without chromatic alteration, then there is no scale that contains both of these chords. Below, I will show you why this is, and what common alterations allow them both.
"...is there an easy trick to determine this?"
Absolutely! Just look at the Circle of Fifths. In the image below, I'm actually using a segment of the infinite Line of Fifths, since it works exactly the same, but will help you to pick the correct spelling for black keys.
This image works like a slide-rule. Beneath the line of fifths are two colored rows: the top for major keys, and the bottom for minor keys. Slide the colored part left or right, along the line of fifths, until the tonic column (I or i) lines up with the desired major/minor key. The roots of all other chords can now be read off the chart. Note that the orange chords (IV, I, and V) are major, the blue chords (ii, vi, and iii) are minor, and the funky yellow chord (vii°) is diminished.
If you're in the minor key (second colored row), the only things that change are the roman numerals associated with the chords (you add 2 or subtract 5), but the pattern of 3 majors, 3 minors, and a diminished stays the same. This gives you VI, III, and VII as major chords; iv, i, and v as minor chords; and ii° as a diminished.
As you can see, there's no way to slide it so that A is minor (blue) and E is major (orange). That means that in order to get both chords, we must chromatically alter one of them. We could pick a key containing an E minor chord, then alter it to be major, or we could pick a key with an A major chord, then alter it to be minor. (In the keys of D major/B minor, we'd have to alter both chords, since we'd have both an A major and an E minor).
However, there is an extremely common technique in the minor key, which is to raise the third of the v chord by a half step, which makes it a major chord (V). Note that the third of the v chord is also the seventh scale degree, so raising this note by a half step places it just under the tonic, creating a "leading tone" which wants to resolve to the tonic. This variant of the scale is called the Harmonic Minor. In the graph, this would be equivalent to replacing the blue "v" in the second row with an orange "V". In this case, you can see that you could have an A minor chord (i) and an E major chord (V) in the same scale: A harmonic minor.
Although the harmonic minor is an extremely common technique, it is not the only possible way to alter chords to get both E and Am together. By moving the colored portion around to different keys that contain both E and A, you can find what alterations are required to get them both. Here's some alternatives:
In C major, you'd be replacing iii with III, which causes it to act as a secondary dominant of vi (V/vi). This is quite common to do.
In G major, you'd be replacing vi with VI, which causes it to act as a secondary dominant of ii (V/ii). This is a bit less common.
In D major and D minor, you'd be replacing ii (or ii°) with II, which causes it to act as a secondary dominant of v (V/v). You'd also need to replace the V chord (which occurs naturally in D major, or in the altered D harmonic minor) with v. Making a dominant chord minor is a less common technique (at least in classical music), and it has the effect of weakening the sense of tonality, and increasing the sense of modality.
In E minor, you'd be replacing i with I (making the tonic a major chord), which might be used as a "Picardy Third" at the end of a piece, or to modulate to the parallel major.
In A major, you'd be replacing the I with a i (the reverse of the above) which can be used to modulate to the parallel minor.
In E major, you'd be replacing IV with iv. This minor subdominant tends to have a melancholy sound to it, and is good in chromatic passages leading up to a cadence. This is another common technique.