It really doesn't matter as neither one will affect the other. Most inexpensive carpets have a polypropylene backing, and that polypropylene won't soften if it gets wet because it doesn't absorb moisture. Good quality carpets will have a latex backing that won't allow liquids to seep through the carpet to cause stains or damage in the concrete or wood below, and that latex won't be affected by shampoo'ing the carpet either.

But, if it were me, I would stretch the carpet first, and then VACCUUM it, and then shampoo it.

That way:
1.) you're not trying to shampoo a wrinkled carpet, so the work will go easier
2.) you're not kneeling down on a damp carpet when you're stretching it, and
3.) the vaccuum cleaner removes all the SOLID dirt (like skin cells, pollen, paper fibers and such) from the carpet before that stuff gets wet from shampooing. Once the carpet and the solid dirt are wet, you'll have much greater difficulty removing that solid dirt cuz of water's surface tension. Next time you're at the beach, try removing sand from wet feet and dry feet and see which one works better. It's the same with carpet. The water film on the carpet pile makes the solid soils stick like glue to the carpet pile.

Contrary to popular belief, a carpet shampoo'er doesn't clean a carpet better than a vaccuum cleaner. They are two different tools meant for removing two different kinds of soils. You need to vaccuum regularily to remove the solid soils, and shampoo periodically to remove the dried up liquid spills.

If you don't read anything else in this post, read the following three paragraphs.

Also, if you're planning on renting a carpet shampoo'er to do the shampoo'ing yourself, then use only 1/5th of the carpet shampoo that Rug Doctor or whomever recommends. That's because Rug Doctor (and that other company too, forget their name) recommend using 5 times as much carpet soap as the companies that make the carpet soaps that professionals use. And, if I was a suspicious person, I'd suspect the reason for that is because they know that if you follow their instructions, there'll be a lot of soapy water left in that carpet after you finish shampoo'ing. As the carpet dries, the residual soap will form a film over all the carpet fibers which solid dirts will stick to, causing the carpet to get dirty much faster than it otherwise would, and making regular vaccuum cleaning ineffective at removing that dirt.

Under those conditions, the ONLY thing that will remove that solid dirt stuck to the soap film on the carpet will be to shampoo the carpet again, thereby simply replacing that soap film with a new one. And, when people see how much dirt is being removed by shampoo'ing, and how much cleaner the carpet is, they're impressed with how clean the rental machine is getting the carpet. Truth is, most of that dirt wouldn't have even be in the recovery tank water were it not for the residual soap left behind the last time the carpet was shampoo'ed; it would have been removed by vaccuuming. The only reason it's there is because it stuck to the soap film on the carpet fibers.

But, if you feel compelled to follow Rug Doctor's instructions to the letter, then at least do a SECOND pass over the carpet with just clean hot water in the solution tank. That way, you remove that residual soap film from the carpet so that you break that cycle of having to shampoo all the time to keep the carpet clean.