Next to Impossible. Probably Impossible.
I'm an electrical engineer and I've answered a number of questions about reverse-engineering things on this Stack. , , 
First, I need to challenge a few of your premises
You're suggesting that an object sitting in front of me could (a) have its controls manipulated in a way that opens a wormhole and (b) can be duplicated with existing terrestrial technology while (c) depending on unknown physics.
The problem with (a) is that it's a technology dichotomy. The proverbial time machine that can be turned on by flipping a single switch. That's great for Hollywood, but a quick visit to any nuclear power station (any power station...) would quickly demonstrate that the idea of a simple set of commands bringing to pass a wonderfully complex result simply isn't true. Now, to be fair, it's theoretically possible that a
really bad programmer wrote the code in such a way that a single easy-to-guess command (aka, "go") would turn on the wormhole. In reality, very intentional security and safety protocols exist to stop that very behavior. So, we have premise problem #1: the easy-to-access-magical-result problem.
The problem with (b) is that the physics of the materials we know about are, in fact, quite well understood. While it's theoretically possible that they can be used in a way we've never thought of that has a wondrous result, the reality is that the elements are still composed of electrons, neutrons, and protons, and we have a pretty good idea about how those work. If we make the assumption that there is the possibility of using existing elements in a way that does something not embraced by our current understanding of physics, your question instantly fails. Maybe we figure out the physics, but it would take years (if not centuries) to build the factory infrastructure necessary to replicate the device. People in the 1960s could reverse engineer today's microchip technology, but it would still require 50 years of factory development to build it. So, we have premise problem #2: the we've-been-able-to-do-it-all-along problem.
And then there's (c). I'm a fan of beating up Dark Matter, not that I don't believe Dark Matter exists (I do, ask me why someday, it'll make you laugh), only that it's an easy punching bag for this purpose. You see, nobody has proven that Dark Matter exists. There is no empirical evidence. There are a lot of believers (on the level of religion!), but no proof. At the moment, Dark Matter is a mathematical band-aid that has two possible repercussions: (1) Dark Matter exists and we simply haven't figured out how to see it yet and (2) Dark Matter doesn't exist because our mathematical models are incomplete and we haven't figured out how to improve them yet. Your premise sits on the very sharp edge that Dark Matter sits on. We humans have the ability to observe an effect, but no ability to observe the mechanism causing the effect. So when you ask, "can we reverse engineer it?" I respond with, "we can't prove Dark Matter exists, so probably not."
Next let's get into the reality of reverse engineering
Reverse engineering, for example, the microchips of today is really hard. Not impossible, but hard. You need to very carefully take the packaging apart because the chip is often destroyed during the dismantling process. We can use various forms of X-ray-ish technologies to look inside, but they won't tell you things like how the substrate is doped (identical densities) and may not tell you where all the wiring goes (identical densities). Even if you know how the microchip operates (you know how manipulating the voltages on pins produces predictable results on other pins), you don't know how it's doing that — which is the point of reverse engineering. And all that assumes that one has discrete components (e.g., transistors) on the chip, which is almost never the case. All kinds of tricks are used to create hybrid devices with the purpose of manipulating variables (size on chip, power consumption, speed of operation) to achieve the expected pin limitations and behaviors.
And we know how every bit of that physics works.
Now, let's do this with a device where we don't understand squat
People who are telling you that it's plausible to reverse engineer this device have no experience reverse engineering anything. That's a blunt statement, but it's the truth, and all I have to do to prove it is point at Dark Matter. If it exists, then the probability is that some of it exists right in front of my nose. And yet we can't detect it, manipulate it, or prove its existence other than mathematically — which means our models can be just as easily in error.
More to the point: some years ago my wife and I enjoyed a historical train ride in Ely, Nevada. During the tour of the maintenance facilities I had a chance to talk with some engineers about their efforts to restore trains. They became excited when they heard that I was an electrical engineer that liked trains. Why?
Because they had a device that was used in a particular model of train to regulate various electrical conditions for engine control and they had no honking idea how to replace it. It was the only one that worked and they had multiple trains that needed it. They hoped I could help them reverse engineer the device based solely on the proverbial gazintas and gazoutas. I spoke with them for a couple of hours about it, and in the end they understood the task was nearly impossible. They had inoperable versions that they'd cracked open and that didn't help solve the problem. It appeared that the device was a very complicated hybrid transformer with both inputs and outputs with no ability to discover winding ratios or meaningful tap locations (much less anything else).
Is this a particularly unknown problem? Nope. NASA has had the devil of a time replicating the Saturn V F-1 rocket engines. Once again, they understand all the physics, and yet the engineering tricks used to overcome manufacturing limitations (among other things) that haven't been a problem for decades have been all but lost to history. Result: we humans are having trouble reverse engineering something we built in the first place.
Conclusion: You can if you want it to as the author, but in real life, nope.
Sitting in front of me is an object. It has controls in a language I can't understand and those controls are certainly connected to a computer using the same degree of technology surrounding the effect you want to reverse engineer, which means I can't figure out how to make the computer work, either, much less reverse engineer the software and the peripheral control connections. But let's assume I can do all that.
What I'm left with is the other object sitting in front of me that embodies in every way the very first flow diagram of every complex technological project ever envisioned by man that contains a box with the words, "magic happens here." I don't understand the physics, do I understand the physics of the controls to the device? Probably not. That means that once I've disconnected the "magic happens here" object from the control device, what's left is something I don't know how to manipulate and don't understand how it creates the results.
Having mechanically damaged and electrically blown up chips I was trying to reverse engineer — which included chips I designed (yet another story that might make you laugh)... — I can tell you that from a Real Life perspective it's more likely that reverse engineering a device that can create a stable wormhole will result in a mile-deep crater than the capacity to duplicate it.
If I take my time to avoid the mile-deep-crater scenario, then what you have is mostly likely a slight benefit to the time that would have been required to develop the technology without the example. (C.F. my comment about people in the 60s reverse engineering today's chips.)
However, you've declared that the use, once discovered, needs nothing more than the technology we have today. Since you're handwaving that, you can handwave this. Thus, you need only to declare it to be so and move on with your story. That's what the Star Gate people did. They completely ignored the physics of what they were dealing with and assumed a 5th-grade control interface because in the end, the wormhole was nothing more than a MacGuffin for the stories they really wanted to tell.
Edit: Some people are experiencing a difficulty that I didn't think would be a problem. They're having trouble separating the complexity of a process from the complexity of its control and the complexity of its purpose.
An example would be an automobile. What a car does is very simple. It moves forward and backward. How it rolls forward and backward can be complex: anything from a set of pedals to things like turbines, rockets, electric motors, combustion... It depends on things like transmissions, emission control, brakes and steering. The important point here is that it's easy to "reverse engineer" the behavior of the automobile because what an automobile does is simple. But that isn't "reverse engineering." All that really is, is duplicating a behavior. I can do that by walking.
If all I cared about was duplicating my competitor's new whiz-bang CPU, all that matters is the very simple and highly predictable behaviors associated with the pins of the microchip. How the CPU works on the inside becomes irrelevant. But this isn't reverse engineering, and it ignores the reason people reverse engineer things.
I attended a lecture in college given by engineers from a business that built Field Programmable Gate Arrays (FPGA). Simplistically, an FPGA is an array of NAND logic gates that the user can configure to perform fairly complex logic behavior. One of the benefits of their design was that all unused NAND gates were disabled as part of the configuration process. One of the attending students asked, "why would you bother to do that?" I've never forgotten his answer.
"Because if we don't, our competitors will."
Why is that lesson important? Because the result of disabling the unused logic gates means the component will use less power, resulting in a more economically efficient product in terms of both the cost of powering the device and the cost of cooling the system. That's a reason to reverse engineer a chip! Because if the goal is simply to duplicate the behavior, automatically disabling unused logic gates can be ignored and the result is a part that can be used in the place of the original even if there are inefficiencies that may have consequences.
A Formula-1 race car can be replaced by a Big Wheel. The efficiency will stink, but the Big Wheel will do the same thing the Formula-1 racer will do: roll forward.
I've used such a simple example on purpose. Reverse Engineering is defined as understanding how something does what it does. It is NOT defined as how to use something. Anyone can pull the trigger on a gun. Given the basic parts, most people can assemble a working gun. But I personally know gunsmiths, and your average person cannot design and build a gun. Oh, they might design a tubular object that could pop out a roundish object with enough force to kill someone — but they'd be more likely to kill themselves using it (mile deep crater...). That's part of the point. Especially when the basic understanding of how a gun works already exists.
A stable wormhole is not "rolling forward" and we have no idea how to make one. It's not a gun, or an automobile, or a cell phone, that the world already has a basic understanding of how to use and the ability to pass that knowledge along to even the youngest of the next generation. Unless I've misunderstood the OP, the goal of the question is not to "roll forward," which is all a wormhole does. It's nothing more than a moving sidewalk, a rickshaw, a means of transportation. In its purpose, it's no different from walking.
But how it completes its function is what I believe the OP is asking about. A stable wormhole is a highly complex consequence of something the OP claims we don't understand at all — and I'm taking the OP at his/her word that the goal is to reverse engineer whatever that something is. Not simply to use it, but to actually understand and replicate it in every way. If that's not what the OP is looking for, I'm willing to be corrected... by the OP.