The connector is made by JST, as the linked page says. I know from looking at it and using lots of JST connectors that it's PH series connector, but you can also find that by looking at this Sparkfun page, or by (tediously) looking through Digikey and JST catalogs.
The plastic part of the connector you linked is called a "housing", and it holds the crimp terminals that are crimped onto the wire, and which slide over the housing. That one is part number PHR-3 (PH series, rectangular (1x3), three terminal). You can get it from Digikey here, or from Future Electronics. Mouser isn't a JST distributor.
However, you'll definitely need the terminals that go with it. They're P/N SPH-002T-P0.5S for the 24-30 gauge wire. (You'll also need stranded wire for this. The picture is nice and big on the Digikey site, but the terminal is TINY.
Unless you have this $400+ tool, you cannot effectively crimp these terminals. Note that I've sucessfully crimped these terminals with a PA series crimper, if you happen to have one of those on hand... You need to strip as little insulation as possible (The long tabs grip the insulation and provide strain relief, while the short ones grip the wire), a good hand on your soldering iron, and plenty of flux on both the terminal and the wire. It can be done.
What you need instead is a KR-series connector. They're IDC style, (Insulation displacement connector) so you don't need to do any crimping. The datasheet is here:
and you'll easily see from the datasheet that what you want has the part number 03KR-6H-P. Unfortunately, Digikey doesn't keep these on hand. Finding them is left as an exercise to the reader.
All that being said, I've also used the Sharp IR distance sensors before I had access to the crimp tool at my current place of employment, and simply desoldered the existing connector, soldered on my own wires, and hot-glued the assembly to prevent strain on the wires.