The Farfallino Alphabet
Japt, v2.0a0, 7 bytes
r\v_²íf

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Explanation
r           :Replace
 \v         :/[AEIOU]/gi
   _        :Pass each match through a function
    ²       :  Duplicate
     íf     :  Interleave &quot;f&quot;


Note: This solution was posted before the &quot;bonus&quot; of handling double vowels was made a requirement and before the (apparent) requirement that when repeating an uppercase vowel the second occurrence should be lowercased. I don't much feel like wasting time on a challenge whose spec is constantly changing and invalidating the solutions that have already been posted, nor do I feel I should be expected to sacrifice 90 rep by deleting a solution that was valid at the time of posting.