Increasing space between chromatics in key signatures and individual chromatics

Using Inspector?
If it is sufficient for your needs (I'm afraid not)

or workaround

Normally you should not attempt to deviate from standard notation practice, so it is best not to mess with with the default settings unless you are trying to create some sort of experimental notation (opening up space for additional symbols, perhaps).

That said, there are settings in Style / General / Notes that can help control the spacing between accidentals and between the accidentals and the note to which they are attached. The rules for accidental placement are quite complex and we actually do a pretty sophisticated job of "kerning" the accidentals so they overlap by appropriate amounts where possible, and thus the settings in the style do not have the direct effect they might if we were more naive. Different music fonts have different "kerning" settings built in to them, and some might call for wider spacing than others.

You can also double click individual accidentals to move them, same as for most other symbols.

Have a look in the Handbook for "Custom Key Signature". You can make your own key signature the way you want it to look. For playback and for automatic handling of accidentals within the score, however, you will need a "real" key signature but you can split the first bar, add the real key signature and hide it. The snag is that you will need to repeat this trick at the beginning of every stave.

What is your use case?
What's wrong with the standard?

As underquark mentions: You can make your own key signature the way you want it to look.

Furthermore, you can save it in a custom palette for re-use, and it will populate all staves if dragged (or double clicked) onto a blank score. Then, if you are transcribing/copying a score strictly for printing or viewing, you can type in notes as you see them. That is, an 'E' is entered as 'E' -- though it will not honor the custom (i.e. fake) key signature for playback.

If you desire playback, you'll have to enter all notes into a real key signature (in this case, Eb), then select all the Eb key signatures and double click on the custom key signature in the palette to populate all staves with the custom key signature. At this point, the accidentals will show (because of the fake key signature); however, you can make them invisible - so as to 'honor' the fake key signature.

See mscz attachment.

Regards.