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<-s%in%c("a","e","i","o","u")]);cat(s,sep="")

Jelly, 15 bytes

f€“¡ẎṢɱ»ðœpżFẊ¥

Try it online!

How it works

f€“¡ẎṢɱ»ðœpżFẊ¥  Main link. Argument: s (string)

  “¡ẎṢɱ»         Yield "aeuoi"; concatenate "a" with the dictionary word "euoi".
f€               Filter each character in s by presence in "aeuoi".
                 This yields A, an array of singleton and empty strings.
        ð        Begin a new, dyadic chain. Left argument: A. Right argument: s
         œp      Partition s at truthy values (singleton strings of vowels) in A.
            FẊ¥  Flatten and shuffle A. This yields a permutation of the vowels.
           ż     Zip the partition of consonants with the shuffled vowels.

R, 99 98 89 bytes

x=el(strsplit(readline(),""))
z=grepl("[aeiou]",x)
x[z]=x[sample(which(z))]
cat(x,sep="")

Seems to be the first human-readable solution! Thanks to Giuseppe for saving 9 bytes!

Test cases:

tho qaeck bruwn fux jemps over tho lozy dig.
progremmang pozzlos & cide gulf

Seems that there is no way to make an internal variable assignment (inside, like, cat), and again some people are going to prove I am wrong...