I'd say they wouldn't be able to, not without overcoming a number of technical barriers first.
So you have 1 scientist who knows how weapons and vehicles work, and some schematics- I'd say they were a long way from being able to implement these, depending on what the scientist's background is (does he have a background in chemistry, as well as more practical aspects such as machining, engineering etc or is it more designing things on the drawing board and letting others do the fine details).
Firstly, in order to make the weapons, you need the raw resources. In order to produce metals capable of containing the pressure of firearm propellants, you're going to need a lot of trace minerals (elements like manganese were used quite a bit in the 20th century firearm production, for example). You're going to need things like chromium and tungsten for the machine tools to make the components. You're also going to need a lot of knowledge like how much trace elements are needed, how to heat treat the materials, etc. You're also going to need to know how to mine and refine these resources- all of
which seems like it's going to be outside of the scientist's expertise, although a civil war society may have some understanding of these processes.
Once you have the actually raw materials like steel produced, you then have to manufacture the parts. A civil war tech society may have some exposure to milling, however it would be a fairly immature technology if they have even invented it yet, and there would be a lot of kinks to work out. Techniques such as stamping sheet metal would be used pretty extensively by the Bunkerites, and as this is an early 20th century technology the USCA would have absolutely no experience with this, this would make mass manufacturing very difficult as everything has to be milled. (One real world example I can point to is the Soviet Union, trying to adopt technologies learned from the Stg-44 after WW2. Even with help from German small arms designers, there was a number of manufacturing hurdles they had to overcome.)
Tolerances (i.e. how accurately you can machine something), are going to be vital in manufacturing weapons and vehicles. Improving this is going to be one of the scientist's key tasks.
Trying to reverse engineer modern weapons is not going to be an easy challenge, and unlikely at best. However, on writing this, one thing that could work is the scientist helping the USCA develop WW1 / interwar level weapons. A civil war tech society is probably on the route to developing firearm cartridges, and maybe 20 years or so from developing smokeless powder (which is something the scientist is likely well suited to assisting them with.) The USCA may have tried developing weapons like black powder revolvers, paper cartidges etc. Once integrated cartridges, smokeless powder and better manufacturing tolerances have been achieved, it isn't too much of a stretch to begin to work on bolt action rifles, and semi automatic handguns, as well as heavier machine guns (all probably 30 ish years away for a civil war tech society).
As a significant amount of firearms development has been collective learning on how to approach certain problems (like the best gas systems for semi automatic weapons, how to develop magazines, what propellants to use etc), the scientist has a lot of hindsight to apply to these problems. It would probably take a number of years to get working prototypes out for bolt action rifles, handguns and machine guns, but it would definitely be possible to massively improve the capability of the USCA given where they already stand technologically.
The scientist would also be able to help developing internal combustion engines, however again trying to jump all the way to modern tech levels would be very difficult.
Edit: while not 100% relevant to your question, this video analyzing an attempt at reverse engineering the 1911 pistol by a Vietnamese craftsman does give some interesting insights to the issue.