The OP asked how to place the script in the .vimrc. This gets just a little tricky because of the odd way Vim import files do line continuation. It would be something like this:
command Typeset call Typeset()
fun Typeset()
let $TYPESET_PATH = expand("%:p")
let $TYPESET_ROOT = expand("%:p:r")
let $TYPESET_DIR = expand("%:p:h")
!sh -icx
cd "${TYPESET_DIR}"
; lilypondPDFoutput="${TYPESET_ROOT}.pdf"
; /Applications/Lilypond.app/Contents/Resources/bin/lilypond -dno-point-and-click "$TYPESET_PATH"
; wait
; open "${lilypondPDFoutput}"
endfun
Here s what is actually now working for me in quite a different environment (Lilypond/Win32; Vim for Cygwin).
" Quick compile command for Lilypond.
command Typeset call Typeset()
fun Typeset()
let $TS_NAME = expand("%:t")
let $TS_DIR = expand("%:p:h")
let $TS_PDF = expand("%:t:r") . ".pdf"
!sh -icx cd "${TS_DIR}" && lilypond "${TS_NAME}" && cygstart "${TS_PDF}"
endfun
Notes: Lilypond/Win32 does not understand forward-slash paths. Therefore I eliminated the path in its argument. You could do the same. You have already set the path with "cd". Also for my environment I took out the point and click option, as well as the "wait", and changed "open" to "cygstart". At that point the shell part was short enough I did not need the rather arcane line continuation demanded by Vim. At the same time I added shortcut operators so that an error at any stage would stop the process.