How to Trim Number from Variable in AppleScript

You're embedding RandomNumA inside a literal (double-quoted) string, so it's treated as part of the literal rather than as a variable reference. To get it interpreted as a variable, you have to do the same thing you do in the do shell script command at the end -- use & to append a quoted literal, a variable reference, and another quoted literal. Actually, it's slightly more complicated because you have to use parentheses to make sure the whole thing gets quoted form of applied to it:

set RandomNumA to (random number from 0 to 65535)

set TurnOnBulbA to the quoted form of ("{\"on\": true,\"hue\": " & RandomNumA & ", \"sat\": 240,\"bri\": 90}")

Here is a simple handler that, pads based on your question.

Example AppleScript code:

set theNumber to 1

set theNumber to padNumberAsString(theNumber)

on padNumberAsString(n)
    set theLength to (get length of (n as string))
    if theLength is equal to 1 then
        set n to "000" & n
    else if theLength is equal to 2 then
        set n to "00" & n
    else if theLength is equal to 3 then
        set n to "0" & n
    end if
    return n
end padNumberAsString

Result: "0001"

As a demo, this creates and displays a list of the padded numbers between 0 and 1000 to show the padding is taking place.

set thePaddedList to {}
repeat with i from 0 to 1000
    set end of thePaddedList to padNumberAsString(i)
end repeat
choose from list thePaddedList

on padNumberAsString(n)
    set theLength to (get length of (n as string))
    if theLength is equal to 1 then
        set n to "000" & n
    else if theLength is equal to 2 then
        set n to "00" & n
    else if theLength is equal to 3 then
        set n to "0" & n
    end if
    return n
end padNumberAsString