R, 57 bytes

function(s,t=s%in%utf8ToInt("aeiou"))s-t*32*!cumsum(t)%%2

Try it online!

Input is vector of character codes (for comparison to pajonk's answer). Would be 77 bytes with input as string (for comparison to Maël's answer).

R, 57 bytes

function(s,t=s%in%utf8ToInt("aeiou"))s-t*32*!cumsum(t)%%2

Try it online!

Input is vector of character codes (for comparison to pajonk's answer). Would be 77 bytes with input as string (for comparison to Maël's answer).

Edit: this approach would be only 48 bytes by porting dingledooper's great modulo trick...

R, 57 bytes

function(s,t=s%in%utf8ToInt("aeiou"))s-t*32*!cumsum(t)%%2

Try it online!

Input is vector of character codes (for comparison to pajonk's answer). Would be 72 bytes with input as string (for comparison to Maël's answer).

R, 57 bytes

function(s,t=s%in%utf8ToInt("aeiou"))s-t*32*!cumsum(t)%%2

Try it online!

Input is vector of character codes (for comparison to pajonk's answer). Would be 77 bytes with input as string (for comparison to Maël's answer).