Abba: The vowels in their name stood for vowels in the group members' names as well.
The Velvet Underground: Introduced zeugma into rock lyrics.
The Monkees: I think the "y" in monkey is voiced like a consonant, don't you? Yuh. Monkey-yuh. Right? So they picked up a vowel here.
The The: Still cracks me up! The The. The The The The The. The.
Van Der Graaf Generator: The doubled "a" is surprisingly common among Dutch bands, but is eschewed by Frisians.
Van Morrison: Not actually a band, but a person. His real name is Van.
Ebn-Ozn: Lost two of their vowels in a lawsuit, sparking their satiric masterpiece "AEIOU and Sometimes Y".
Styx. Name doesn't look like it has any vowels, yet it does. Look closely.
AEIOU: The only band name to have all the standard vowels of the alphabet, in alphabetical order!
XTC. Technically not qualified for the list, but I didn't think I could leave them off.