: E major and A minor together are not in scale? I was making up a chord progression on guitar which is F, E, Am, C. I then attempted to find what key this progression could be in and I'm getting confused because I cannot

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Because there are more than one minor scale, it is a fact that E major is part of the minor 'system'.

The online tool appears (I've not seen it) to address only the natural minor. And it's correct - so far!

The other common minor scales (no point in getting involved with modes in this case) are melodic and harmonic, both of which contain, in A minor, the G# note. Thus, making chords (triads) from those scales, both of them, will produce E major, and even E7.

One of the reasons these two minors include G# is that it is now a proper leading note, a semitone away from the tonic, giving the sound a more powerful pull from G# to A, than it otherwise would from G to A.

So, all these chords are seen to belong to A minor. And, yes, E minor can be used as well in A minor key, although usually, a piece will stick to one or the other. For future reference, these notes, and the chords they spawn, are in 'A minor': A, B, C, D, E, F, F#, G, G#. And in case it's deemed necessary to also use those of the parallel key - C# lurks within...

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It helps when trying to find a scale to fit a series of chords to make a list of their notes:

F - F, A, C
E - E, G#, B
Am - A, C, E
C - C, E, G

A couple of things jump out at me here. First, you have 3 notes clustered with only semitones in between: G, G#, A. While it is entirely possible to find scales that have 3 half steps in a row (like bhairav), this can make things a little tricky. The second thing that occurs to me is that the only sharp in there is the G#.

If you look at the set of notes you end up with, it looks a lot like a mode of C except for that G# and the fact that there's no D. You should feel free to use a D in a guitar solo or melody I think and wouldn't sound too unnatural.

A, B, C, E, F, G, G#

I can't name what rules I'm using to to come to this conclusion, but that looks a LOT like A minor to me -- and if you need a scale, consider using an A minor scale and play G when descending and G# when moving up the scale. If you're doing a solo or something like that, the B/C and E/F points should make for interesting focal points. While this might not be the canonical A minor scale, i.e., the "natural" minor scale, also known as the Aeolian mode of C, it is quite common for songs written in A minor to shift that G to G# for two good reasons:

you get a nice tasty dominant V chord (E G# B) which resolves strongly to the root i chord (A C E). Throw in that missing D for an E7 chord.
that G# yearns to become your root A note and resolves very strongly to it.

The fact that you start on the F chord suggests something a little weird (simple progressions typically start on the root of a scale) but not especially so.

For future reference, I might suggest using this scale finder. Entering all those notes from your chords, it comes back with nothing, but if you leave out the G it comes back with A harmonic minor. If you leave out the G# instead, it comes back with a whole bunch of possible scales.

All in all, I'd say you are in pretty solid A minor territory. The wikipedia article on minor scales has some fairly helpful detail about harmonic minor scales and melodic minor scales which might morph a bit depending on whether you are going up or down. In practice, composers have done all kinds of unorthodox things when using minor scales. Part of the fun of minor scales is that they embrace dissonance and encourage breaking the rules.