AppleScript do shell will not work on Application Support directory

If you are hardcoding, without testing if the file exists, then use the following example:

set foo to quoted form of "/Users/vader/someapp.app/Contents/Resources/Scripts/foobar.txt"
set bar to quoted form of "/Library/Application Support/"
try
    do shell script "cp " & foo & space & bar with administrator privileges
end try

If you are calling it from the script within the application bundle, then use the following example:

on run
    try
        set foo to POSIX path of (path to me as string) & "Contents/Resources/Scripts/foobar.txt"
        set bar to quoted form of "/Library/Application Support/"

        tell application "System Events"
            if exists file foo then
                tell current application
                    do shell script "cp " & quoted form of foo & space & bar with administrator privileges
                end tell
            end if
        end tell
    on error eStr number eNum
        display dialog eStr & " number " & eNum buttons {"OK"} default button 1 with icon caution
        return
    end try
end run

Note: When working with the do shell script command, POSIX pathnames must be used, not aliases. Macintosh HD is not valid in a POSIX path, just in an alias. In a POSIX path of the startup disk, everything starts in the root of the volume: /

Example:

tell application "Finder" to get name of startup disk as alias
tell application "Finder" to get POSIX path of (name of startup disk as alias)

Returns:

Macintosh HD 
/

@user3439894 has already explained how to solve the problem; I'd like to take a stab at explaining why the confusion exists. It goes back to the fact that the Macintosh OS wasn't always based on unix (/POSIX). Back before OS X (which is unix-based), Mac OS was a completely different OS with its own file path notation. When OS X came out it had to support both the unix file access APIs (which used unix path notation) and the traditional Mac OS APIs (which used traditional Mac OS path notation). Over the years, OS X (/macOS) has deprecated some of the traditional Mac OS APIs and switched more and more towards pure unix path notation. But AppleScript is an exception: it predates OS X, so it natively uses the old traditional Mac OS notation.

Here's a summary of the differences:

• Unix (/POSIX) paths use "/" as delimiters (and allow ":" within filenames), while Mac OS paths use ":" as delimiters (and allow "/" within filenames).
• Unix paths are absolute if they start with the delimiter ("/"); Mac OS paths are absolute if they do not start with the delimiter (":").
• Absolute Mac OS paths start with the volume name, while absolute unix paths start at the top of the root volume, and other volumes appear as folders somewhere under it.

Here are some examples:

Mac OS: Macintosh HD:Users:vader:someapp.app:Contents:Resources:Scripts:foobar.txt
Unix: /Users/vader/someapp.app/Contents/Resources/Scripts/foobar.txt

Mac OS: Other disk:daily notes:7/19/2017 meeting.rtf
Unix: /Volumes/Other disk/daily notes/7:19:2017 meeting.rtf

Mac OS: :Documents:file.txt
Unix: Documents/file.txt

In AppleScript, you can use POSIX path of and as alias to convert between these.