Yes, there is such thing as the "key of E Mixolydian", but it's not a proper way to address it. You should call it the "mode of E Mixolydian". Notating it on sheet music is more complex then just putting the key signature of A major/ F♯ minor.
First off, keys and modes are not the same thing which is why use Major and Minor to represent keys rather than Ionian and Aeolian. Modes predate the idea of keys and mixing the concepts leads to a lot of confusion since most modes don't follow the traditional tonal concepts of harmony so calling them keys doesn't make sense. So semantically there is no "key of E Mixolydian" nor "key of C Ionian" for that matter you should refer to them as modes instead.
However, the concept of a key in general especially with a more modern eye and less emphasis on traditional functional harmony opens up a lot of possibilities.
A key gives you two pieces of information, it tells you the tonal center of a piece and the expected harmony and melody of the piece. Again the tonal center is more for tonal ideas, but it can easily be extended to modes. The harmony of Mixolydian vs Major is also vastly different so saying something is specifically in the mode or key of Mixolydian is valuable information.
How to represent it on sheet music however is not as straightforward. Let's take E Mixolydian as an example. You could notate the key signature two ways which are:
- You can write the key signature for A major/ F♯ minor and hope people understand that you're in the mode of E Mixolydian.
- You could write the key signature for E major to demonstrate the tonic and use
A real example of a song in E Mixolydian is Good Times, Bad Times which you will find sheet music for both ways.
The issue with the first is musicians have been trained to think the key signature of A major/ F♯ minor will have the tonic be either A or F♯ which is not the case. The signature fits better and represents the set of notes, but information about the tonic may not be clear.
With E major it's much more clear that E will be tonic and since it's a major mode only a few alterations will be needed. However you'll have to introduce accidentals to cancel out the D♯ where as in A major/ F♯ minor you would not have to.
It's up to you at the end of the day.