Modern Vespa

I have a ~2010 Vespa LX50. Gone 13 000 km. I have noticed that the back wheels wiggles. It is loose. Should i be worried about this?

I have attached a video of what I'm describing.

https://drive.google.com/file/d/1yYNT8939n8HMFd-2FE9QLHoDZU3rKwgd/view?usp=sharing

I would be worried about that.

You need to take the exhaust off then the wheel to investigate.
The rear wheel should have a split pin, cover, nut and a thick washer holding it on, it should be done up between 137-152 NM.
It's possible your wheel is damaged somehow but I'd check it isn't just loose first.

holy f. there's no way I would ride that.

You should be able to get your hand in there to check the nut, cotter pin and hold cap without removing the exhaust. Pull the cotter pin and hold cap, see if the nut is loose. Do you have all these parts?

The bearings look to be shot in addition to all the above.

Without a doubt bearing and given age of the scoot it done well, easy to replace, just freeze it first, and a bit of grease and bearing should drop in easily, or use a socket same width of bearing put it on top and tap lightly down with a mallet.