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You can find the first Vowelburger™ Riddle here

Local Vowelburger™ business has been booming lately, which has been great for our CEO's retirement plans. Unfortunately, all those profits have attracted the attention of some ne'er-do-wells. I was discussing plans with my assistant Iva when we heard a loud thud, a smattering of curses, a cat's yowl, and the revving of tires. We rushed to the back of the store, only to discover that our new burgers and recipes had been completely nabbed! Although we managed to get the license plate of the perpetrator (A179374), the cops informed us that the number was bogus and wasn't registered in any plate database.

We must find those Vowelburgers™ immediately, to vet them for quality assurance purposes. As our newest and brightest recruit (with a penchant for solving the unsolvable), we're tasking you with finding the missing Vowelburgers™, as well as figuring out where the thief is heading with those Vowelburgers™. Here are the clues the police discovered upon forensic analysis of the places where each Vowelburger™ went missing:

Meat Clue
A Relative's gout needs no introduction (4)
E Thin stick as end to wood (4)
I Snide wit; pert flower (8)
O Leaving, we eyed game (3)
U Sting underwater professional with point, crushing his head (9)

You're our last hope - please, help us find our Vowelburgers™!

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  • $\begingroup$ Would you like to clarify whether the vowelburgers themselves are regular vowelburgers of the usual pattern? $\endgroup$ – Gareth McCaughan♦ Mar 11 '20 at 13:17
  • $\begingroup$ They look like regular Vowelburgers™, but might not be words per se. It's dangerous to let such defective products out into the public: therefore, it's of utmost importance that we find them as soon as possible! $\endgroup$ – Avi Mar 11 '20 at 13:19
  • $\begingroup$ Ah, I wondered about that. (Not because I've solved any part of the puzzle, but my guess at how it works made it more likely.) $\endgroup$ – Gareth McCaughan♦ Mar 11 '20 at 13:20
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Your burgers have buns

H and N buns

and what has happened is

that they have been removed from the clues, which should read as follows:
Relative's HANgout needs no introduction -> (-h)AUNT.
Thin stick asHEN end to wood -> WAN+D.
Snide witHIN pert flower -> PER(MEAN)T.
Leaving we HONeyed game -> S(-we)ET.
SHUNting underwater professional with point, crushing his head -> DIVER+T(-h)ING.

And now, rather arbitrarily to my mind but no matter,

index into these words using the first five numbers in OEIS sequence A179374 as per that license plate, to find that the thief is bringing the burgers on A DATE. I'm not sure what we can really do with that information, but no matter :-).

Notes:

I initially had a worse interpretation of the wordplay in the first clue; Avi set me right in TSL chat. And I didn't solve the third one; James Coyle, in comments below, did. Thanks, James! Nor did I figure out how to extract the final bit from the words we found; that was hagfy in comments. Thanks, hagfy!

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    $\begingroup$ I think you're looking for rot13(crezrnag) which is a different meaning of flower. Snide -> rot13(zrna) within pert. $\endgroup$ – James Coyle Mar 11 '20 at 14:31
  • $\begingroup$ Yeah, must be right. I don't like it much (using PERT literally, and it's a bit of a strained definition) but clearly correct. Thanks! $\endgroup$ – Gareth McCaughan♦ Mar 11 '20 at 14:39
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    $\begingroup$ @GarethMcCaughan Not sure this is helpful at all, but if you do something like rot13(gnxr gur svefg svir qvtvgf bs gur brvf frdhrapr naq hfr gurz ntnvafg rnpu jbeq, znlor ur'f oevatvat gubfr ohetref gb "N QNGR")? $\endgroup$ – hagfy Mar 11 '20 at 16:39
  • $\begingroup$ That's certainly possible. It doesn't make much sense "in-story", for all sorts of reasons, and it's kinda arbitrary, but these extraction processes often are. @Avi, would you care to comment? $\endgroup$ – Gareth McCaughan♦ Mar 11 '20 at 16:44
  • $\begingroup$ @GarethMcCaughan That's it! $\endgroup$ – Avi Mar 11 '20 at 16:50

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