Anybody know where I can get 13in x 3in alloy rims.

3" width has got to be motorcycle type size?

13" diameter seems too small for a motorcycle though (moped perhaps).

Do they need to be single side attaching like car wheels (not double side like many motorcycles)?

Yep, for maxi scooter tyres.
Chasing a single sided mounting, like a car, as I don't want to support the Scooter wheels spindle of 11mm on both sides with a loop. Ugly!!

I understand the desire for single sided mounting. When I built a sopbax cart I used 26" MTB front wheels for the rear wheels and 24" wheelchair wheels for the front.

Rear weehls supported both sides, front wheel just one side.

The spindle on the scooter wheels is 11mm diameter. I could replace the normal bearings with taper roller bearings and use a single castle nut to locate. But 11mm is a bit too small for an end mounted spindle. Unless I can find a bit id metal strong enough to stand forces generated when doing 50-60 MPH.

In my dim and distant past I rode a Lambretta scooter and IIRC both front and rear wheels were mounted on a single side. No idea what the wheel size was though.

Have you looked at the Piaggio MP3 front wheels - 2 front wheels with single sided mounting and some of them are 13"?

eg

https://www.ebay.co.uk/itm/196136612270?mkcid=16&mkevt=1&mkrid=711-127632-2357-0&ssspo=CYy4rnTJQ-K&sssrc=2047675&ssuid=v7qJTm rGQx6&widget_ver=artemis&media=COPY