Phil's solution worked for me. I wanted to change the line colour as well, so I used:
sudo fswebcam --banner-colour: \#000000 --line-colour \#ffffff --png 9 -r 640x480 /var/www/image.jpg
That worked fine on its own, but when I tried to combine it with watch, I was encountering the same error, despite the hashtags being escaped.
After some Googling, I was able to fix that problem by putting double quotes around all of the code that was being given to watch.
Here's the final code that I used:
watch -n10 "sudo fswebcam --banner-colour: \#000000 --line-colour \#ffffff --png 9 -r 640x480 /var/www/image.jpg"