Fairly easy to do via a webviewer scrape or Insert From URL.
HTML5 has a canvas function called measureText that specifically does this:
If you were to send your filemaker string to a webviewer, you could scrape the result back out.
Or possibly set this up as a web service that sends the result back via Insert From URL, EG:
Set Variable [ $string ; GetAsURLEncoded(table::stringField) ]
Insert From URL [ table::resultField ; "http://www.yoursite.com/length.php?str=" & $string ]
Then on the PHP side you would get the value of str and echo the character result. There is a PHP function that generally does the same:
Thank you Mike for the suggestions and the helpful links. I will see if I can get that working.