How do I automate clicking a button in an application's window with AppleScript?

If the button is not available directly, you might have to guess.

What works in High Sierra is:

tell application "System Events"
    tell process "Image Capture"
        click button "Scan" of group 2 of splitter group 1 of window "Image Capture"
    end tell
end tell

On older macOS releases, group 1 may have to be used.

Another workaround would be to activate the window and press the space bar:

tell application "Image Capture"
    activate
    tell application "System Events" to key code 49
end tell

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You can actually try to near it down by running commands like:

UI elements of window 1
UI elements of splitter group 1 of window 1
UI elements of group 1 of splitter group 1 of window 1

or, as @Lri said in the comments:

properties of UI elements of window 1
properties of UI elements of UI elements of window 1

This will present you with a list of elements contained, and you can guess your way from there.