Short answer
No, EU citizens do not need a visa to enter the Azores, but they need a valid ID card or passport. The Azores are part of Portugal and the Schengen area, so EU freedom of movement applies. For stays longer than three months, you must register with the local authorities. Rules can change, so check the official EU and Portuguese government websites before travelling.
General information, not legal or tax advice. Rules and amounts change: confirm with official sources.