APL 30 2726
,('AEIOUY'∪⎕A'EIOUY'∪⎕A)[6(|,+)⍪5?20]
Explanation is very similar to the past version below, just reordered a bit to golf the solution.
Note: ⎕IO is set to 0
(∪'AEIOUY',⎕A'EIOUY'∪⎕A)[,6(|,(⍪+))5?20]
Explanation:
'AEIOUY'∪⎕A'EIOUY'∪⎕A puts vowels in front of all letters.
5?20 for the indexes we start choosing 5 random numbers between 0 and 19
6(|,(⍪+)) then we sum 6 and the random numbers, convert to 5x1 matrix (⍪), add a column before this one containing 6 modulo the random numbers.
[[[ this one can be rewritten as: (6|n) , ⍪(6+n) for easier understanding]]]
,6(|,(⍪+))5?20 the leading comma just converts the matrix to a vector, mixing the vowel and consonants indexes.