Original Wii (RVL-001 / RVL-101): No HDMI. Outputs:
Wii U (WUP-001): Yes, native HDMI 1080p. Also backward compatible with Wii games (upscaled to 1080p via HDMI).
For pure Wii on modern TV: Wii U or WiiDual mod. Cheap: Component Cable + TV Component In or Wii2HDMI.