Is it actually possible to send food to a lab to get the recipe?

As long as you have a list of potential ingredients, it would be possible to find out if these ingredients are in the sauce. For example, if you don't know what spices were used, you could start with a list of spices, find information on some signature chemical compounds found in each spice of the list, then tell the lab to find out which of these substances are present in the sauce. This would give you a pretty good list of substances (spices) you could use. You will need a curious chemist-food scientist who has experience with that kind of work and is willing to play a detective, I'm pretty sure you can't get to a general purpose organic chemistry lab and expect them to just plug it in and get a result.

Even that information won't be 100% certain. First, you would have to find substances which are present in one source ingredient but not another - and these are unlikely to be the main aromatic substances, since these tend to be shared between plants, for example eugenol is something you'll find in a lot of herbs. Second, you might have unusual combinations in which the plant may get into the recipe: for example, where the lab suspects the use of inverted sugar and hyssop as an herb, it might turn out that the recipe contained wildflower honey and the bees processed lots of hyssop.

And when you get the information, you still don't have a recipe. Both the ratio and the process are missing. A good cook (or food technician, for industrially produced food) can make educated guesses about possible processes, and with some work, they are likely to create some kind of replica, if the original recipe doesn't include surprising tricks.

So, you decide to do so, it is kinda possible, but it will be a long process involving experts, not a send-the-sample-get-full-answer kind of thing. If there are no businesses offering it as a service, it doesn't seem like a practical proposition.