Can Shanghai Social Security buy a car in Jiangsu?

It depends on buying a car. Some high-end brands can have more relaxed regional restrictions, but mainstream home brands generally won't sell them to you. If you have a license, you can't do it yet. Shanghai people have social security restrictions on buying a car in Suzhou. If you don't pay a certain number of years, you can't get a license, or if you have a company in Suzhou, you can provide an organization code certificate to get a company license.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.