CJam, 29 bytes

"aeiou""eioua"l2$eu2$euer@@er

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Uses er builtin.

"aeiou"                         # push lowercase vowels
                                #       stack = [ "aeiou" ]
       "eioua"                  # push rotated lowercase vowels
                                #       stack = [ "aeiou", "eioua" ]
              l                 # push input
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog." ]
               2$               # push copy of stack element 2nd from top
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "aeiou" ]
                 eu             # to uppercase
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "AEIOU" ]
                   2$           # push copy of stack element 2nd from top
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "AEIOU", "eioua" ]
                     eu         # to uppercase
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "AEIOU", "EIOUA" ]
                       er       # transliterate
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog." ]
                         @@     # rotate top 3 items in stack, twice
                                #       stack = [ "The quick brown fox jumped over the lazy dog.", "aeiou", "eioua" ]
                           er   # transliterate
                                #       stack = [ "Thi qaock bruwn fux jamps uvir thi lezy dug." ]

CJam, 29 19 bytes

q"aeioua"_eu+_1m<er

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-10 bytes thanks to @Peter Taylor

q                       # take all input
 "aeioua"               # push vowel pairs
         _eu            # duplicate, uppercase
            +_          # concatenate, duplicate again
              1m<       # rotate left by 1
                 er     # transliterate

Stax, 20 bytes

¡╛▒Æ'├K]&└↔@╛ó┴0*ì╚◘

Run and debug it

Just uses a built-in for translating.

Unpacked:

"AEEIIOOUUAaeeiioouua"|t

CJam, 29 bytes

"aeiou""eioua"l2$eu2$euer@@er

Try it online!

Uses er builtin.

Explanation:

"aeiou"                         # push lowercase vowels
                                #       stack = [ "aeiou" ]
       "eioua"                  # push rotated lowercase vowels
                                #       stack = [ "aeiou", "eioua" ]
              l                 # push input
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog." ]
               2$               # push copy of stack element 2nd from top
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "aeiou" ]
                 eu             # to uppercase
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "AEIOU" ]
                   2$           # push copy of stack element 2nd from top
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "AEIOU", "eioua" ]
                     eu         # to uppercase
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog.", "AEIOU", "EIOUA" ]
                       er       # transliterate
                                #       stack = [ "aeiou", "eioua", "The quick brown fox jumped over the lazy dog." ]
                         @@     # rotate top 3 items in stack, twice
                                #       stack = [ "The quick brown fox jumped over the lazy dog.", "aeiou", "eioua" ]
                           er   # transliterate
                                #       stack = [ "Thi qaock bruwn fux jamps uvir thi lezy dug." ]