How to “require an answer” in a dialog in AppleScript

Probably not.

I would imagine that the lock screen is locked down fairly tight - imagine a piece of malware that did just that - except the Automator action sent the password to a server in Bangladesh.

It can't be done in AppleScript.

However, I did find this workaround, where text entered on each line of one field is interpreted as a separate answer:

-- multiple input dialog

on run -- example
    set {firstName, lastName} to (inputItems for {"• First Name", "• Last Name"} with title given prompt:"Enter the following items separated by a carriage return:")
    display dialog "First Name:  \"" & firstName & "\"" & return & "Last Name:  \"" & lastName & "\""
end run

to inputItems for someItems given title:theTitle, prompt:thePrompt
    (*
    displays a dialog for multiple item entry - a carriage return is used between each input item
    for each item in someItems, a line of text is displayed in the dialog and a line is reserved for the input
        the number of items returned are padded or truncated to match the number of items in someItems
    to fit the size of the dialog, items should be limited in length (~30) and number (~15)  
        parameters -        someItems [list/integer]: a list or count of items to get from the dialog
                        theTitle [boolean/text]: use a default or the given dialog title
                        thePrompt [boolean/text]: use a default or the given prompt text
        returns [list]:     a list of the input items
    *)
    if thePrompt is in {true, false} then -- "with" or "without" prompt
        if thePrompt then
            set thePrompt to "Input the following items:" & return & return -- default
        else
            set thePrompt to ""
        end if
    else -- fix up the prompt a bit
        set thePrompt to thePrompt & return & return
    end if

    if theTitle is in {true, false} then if theTitle then -- "with" or "without" title
        set theTitle to "Multiple Input Dialog" -- default
    else
        set theTitle to ""
    end if

    if class of someItems is integer then -- no item list
        set {theCount, someItems} to {someItems, ""}
        if thePrompt is not "" then set thePrompt to text 1 thru -2 of thePrompt
    else
        set theCount to (count someItems)
    end if
    if theCount is less than 1 then error "inputItems handler:  empty input list"
    set {theItems, theInput} to {{}, {}}

    repeat theCount times -- set the number of lines in the input
        set the end of theInput to ""
    end repeat
    set {tempTID, AppleScript's text item delimiters} to {AppleScript's text item delimiters, return}
    set {someItems, theInput} to {someItems as text, theInput as text}
    set AppleScript's text item delimiters to tempTID

    set theInput to paragraphs of text returned of (display dialog thePrompt & someItems with title theTitle default answer theInput)

    repeat with anItem from 1 to theCount -- pad/truncate entered items
        try
            set the end of theItems to (item anItem of theInput)
        on error
            set the end of theItems to ""
        end try
    end repeat
    return theItems
end inputItems