Ot wes thi bist uf tomis

MS-SQL, 51 Bytes

Works on SQL 2017 or above:

SELECT TRANSLATE(v,'AEIOUaeiou','EIOUAeioua')FROM t

The new function TRANSLATE performs individual character replacement, so is ideally suited for this challenge.

Input is via a pre-existing table t with varchar column v, per our IO rules.

In this case the table must be created using a case-sensitive collation, either by running on a case-sensitive server, or by using the COLLATE keyword (not counted toward character total):

CREATE TABLE t(v varchar(max) COLLATE Latin1_General_CS_AS)

EDIT: SSMS may cut off the lengthy quote above when returning the result in a "results to text" window, this is a client setting, not a bug in my program.

To fix, go to Tools > Options > Query Results > SQL Server > Results to Text and increase the "Maximum number of characters displayed in each column."

Haskell, 52 bytes

(a:b)!c|a/=c=b!c|1>0=b!!0
a!b=b
map("aeiouaAEIOUA"!)

Try it online!

Lynn saved me two bytes by pointing out that !!0 is shorter than head.

Explanation

If you have never coded in Haskell this will probably look like a pile of jibberish. So first let's ungolf it and then break it down:

(a:b)!c
 |   a/=c   = b!c
 |otherwise = b!!0
a!b=b
map("aeiouaAEIOUA"!)

First we have a function !, which takes a string s and a character c. Our first pattern match catches accepts input if the string is non-empty. If the string is non-empty we compare its first character to c. If it's first character is not equal to c we toss it and call ! again with the remainder of the string and c. If it is equal we return the second character in the string.

Our next pattern match catches the string in all other cases, that is if the string is empty. In this case we just return c.

All in all this function takes a character c and a string s and returns the character after the first occurrence of c in s. If we pass this with aeiouaAEIOUA it will perform our cipher on a single character. To make our whole function we ought to map this across the string.

Bash + coreutils, 24

tr aeiouAEIOU eiouaEIOUA

Try it online!