I didn't test it but you need to use :h :norm:
:norm[al][!] {commands} *:norm* *:normal*
Execute Normal mode commands {commands}. This makes
it possible to execute Normal mode commands typed on
the command-line. {commands} are executed like they
are typed. For undo all commands are undone together.
Execution stops when an error is encountered.
Which would give you the following file:
%s/string/string$i/g
norm! V
norm! 19j
norm! y
vs output.file
norm! G
norm! p
wqa
Also note that you don't need to put : in front of your normal mode command when they are written in a script.
Also, for what it's worth I would not use Vim for what you are trying to do. I think grep with the -A parameter would be enough to get only the 19 lines after your matched string. And you can pipe it to sed to do you substitution.
Probably something along the lines of
grep -A 19 "string" file-to-edit | sed 's/string/string$i/' > output.file
If you have many files this should me faster because grep and sed are specialized for these tasks and should be better than Vim. Also since you are now in the terminal you can use the * joker to treat all of your files.