Short answer
US citizens do not need a visa for short tourist stays in the Azores (part of Portugal and the Schengen area): typically up to 90 days within any 180-day period. You need a valid passport. The EU is also introducing an online travel authorisation (ETIAS) for visa-exempt travellers, so check the official start date and rules before you book. Confirm everything with official Schengen or Portuguese embassy sources.
General information, not legal or tax advice. Rules and amounts change: confirm with official sources.