According to my current navigation database (AIRAC 1912), the following 228 airports have a category III ILS approach to at least one of their runways:
CYVR (VANCOUVER INTL) on runways: 08L, 08R, 26L, 26R
CYYC (YYC CALGARY INTL) on runways: 17L, 35R
CYYT (ST JOHNS INTL) on runways: 11, 29
CYYZ (LESTER B PEARSON INT) on runways: 05, 06L
EBBR (BRUSSELS ...
You didn't list a jurisdiction so Ill answer for the FAA: Yes you can continue;
Under part 91 you can both start and fly an approach if the conditions are below minimum.
If you are flying under part 121 then you can't start the approach but § 121.651 Takeoff and landing weather minimums: IFR: All certificate holders. is going to be the regulation to ...