This seems like a perfectly reasonable request. Unfortunately, though, the number of figures available on 1/96 scale (an extremely common one among ship modelers) is surprisingly small.
The ones that seem to show up most frequently are those from a handful of old Revell kits. Revell sculpted a beautiful set of figures for its Cutty Sark back in 1959, and issued them with several other kits: the Thermopylae, Pedro Nunes, Alabama, and Kearsarge. In 1965 Revell did another batch for its new U.S.S. Constitution. (These were completely different figures, wearing 1812-vintage naval uniforms. Be careful, though: there was a period, probably in the seventies, when Revell was packing the old Cutty Sark merchant sailors in Constitution kits. If you find an old Constitution on e-bay or at a flea market, it would be worth checking.) The Constitution got reissued for a while under the name U.S.S. United States.
In the late seventies Revell issued a ... thing ... it called a "Spanish Galleon," allegedly on 1/96 scale. It had a set of figures too. I made the mistake of buying one. When I opened the box and discovered that the contents bore scarcely any resemblance to anything that actually floated, I fear I became slightly catatonic; I don't remember whether the figures were new or recycled from the older kits. (That Revell "Spanish Galleon" doesn't deserve to be labeled a scale model. Best for me to get down off that soapbox at this point.) Revell later did a reissued version called "English Man-o'-War"; I assume it has the same figures.
Anyway, any of those kits would supply plenty of figures - which could be modified and painted to represent Civil War-vintage sailors. The bad news is that (a) most of those kits are hard to find, and (b) if they can be found, they're expensive.
There are a few other plastic ship kits with 1/96-scale figures in them (the old Revell Golden Hind comes to mind), but they're also hard to find and almost as expensive as the ones I just mentioned.
I imagine you've already considered - and ruled out - the various soft plastic Civil War soldier sets made by companies like Airfix, Revell, and Zvezda for wargamers. They're generally on 1/76 scale - a bit big for 1/96.
The German company Preiser makes hundreds of figures for HO-scale railroads. HO, of course, is 1/87. (I think Preiser actually advertises its figures as being on 1/90.) The actual sizes of Preiser figures varies a bit; some of them are in fact 3/4" (i.e., 6' on 1/96 scale) or shorter. With some modification they can be made to work very nicely on ship models - and they don't cost much. (As a matter of fact I think Preiser makes a couple of small sets of Civil War soldiers.) They come in two forms: painted (not very well) and unpainted. The painted ones are sold in sets of five or six figures apiece, for rather high prices (somewhere between $5 and $10, I think). The unpainted ones come in boxes of a hundred or so, and are among the best bargains in the hobby industry. In either case, the place to find them is a well-stocked hobby shop that caters to model railroaders, or a web dealer (e.g., Walthers) that does the same. Might be worth a look.
I believe there's at least one company in continental Europe that makes nautical figures (soft plastic again, I'm afraid) specifically for ship models. I haven't seen any of them in the flesh, but I believe at least one of our European forum members is familiar with them - and a web dealer that sells them. I have no idea whether any of those sets would work for the Civil War. Michel - are you reading this?