Depends on where exactly in the EU you are a citizen. Local rules about tax etc. will apply.

In general, purchasing a vehicle does not require an EU driver's license. You can be the registered owner of a vehicle without having ever had a license - it's a property transaction! You could just give it to a family member to drive, that would be perfectly legal.

Insurance does depend on local rules, because that's a private market transaction - the insurance agency can choose to reject you if you don't have a license, or charge a prohibitively high price. In some countries, insurance attaches to the vehicle and is valid for anyone driving it. In others, it attaches to a specific named driver.

In my country, Estonia, it is perfectly fine and legal for a non-resident to buy and register a car in their name, if they have a local resident who is willing to put their name on the paperwork as the Responsible User. (Basically, the person that the local government calls if they have any questions about the vehicle.) The actual owner doesn't even need to be an EU citizen! I've done this for people who registered their bikes on non-EU passports, and drove all the way to Mongolia and Vladivostok on their Estonian plates, no problem. (Estonia also doesn't have any annual vehicle taxes.)

Finally, you can get a vehicle e.g. in Germany on export plates. These are special plates that mean you will take the vehicle and re-register it outside Germany by the time the plates expire - but before that, you can drive on them freely throughout the EU (or outside). They even come with insurance! And the expiration date can be as long as a year.