Sheffle tho vawols ureund!
R, 92 91

Can't comment yet so I'm adding my own answer albeit very similar to @Andreï Kostyrka answer (believe it or not but came up with it independently).

s=strsplit(readline(),"")[[1]];v=s%in%c("a","e","i","o","u");s[v]=sample(s[v]);cat(s,sep="")


Ungolfed

s=strsplit(readline(),"")[[1]]    # Read input and store as a vector
v=s%in%c("a","e","i","o","u")     # Return TRUE/FALSE vector if vowel
s[v]=sample(s[v])                 # Replace vector if TRUE with a random permutation of vowels
cat(s,sep="")                     # Print concatenated vector




Saved one byte thanks to @Vlo

s=strsplit(readline(),"")[[1]];s[v]=sample(s[v&lt;-s%in%c("a","e","i","o","u")]);cat(s,sep="")