Can you disable the sound when taking pictures?

No. (At least not to my knowledge.)
In many (or most) modern devices you can't turn the photo making sound off.
This is to prevent perverts from taking pictures (of women's panties and stuff) unnoticed.

Which is silly because those that want to do it will find their way of doing it. Same problem with banning guns, criminals don't have problem getting them.

"This is to prevent perverts from taking pictures (of women's panties and stuff) unnoticed."

Yeah, actually a law here in Japan for every device to have this.

So turning the sound off or putting headphones doesn't stop the sound?

It's a law cell phones and other devices must use a sound. Not just to prevent perversion but some companies like car companies or gov buildings would not allow cell phones with cameras since it was easy to take pics of things the should not and leak out.  So devices were then required to have a sound.

Which is stupid, you can record videos without the snapping sound.

It is kinda and maybe needs updating however recoding video is a bit more obvious. Since you have to hold it there for longer period of time.

You could easily buy spycam that disguise as whatever common items...

http://2.bp.blogspot.com/-YJ-Jq0VoR-c/TeXcTS_zhDI/AAAAAAAAAeI/g3GhEodXG3Q/s1600/11397%252Cxcitefun-spy-glasses-1.jpg