How do I find info on whether there a US Army camp at Saratoga (in present-day Omaha) in 1861?
Thanks for asking! The usual suspects for early Omaha-area maps don't have much to offer - the 1866 Gast and Davis maps at least show Saratoga, but do not identify structures in the township. I would encourage you to consult James E. Potter's Standing Firmly by the Flag: Nebraska Territory and the Civil War, 1861-1867. Some very basic information on military jurisdiction in the Nebraska Territory pops up in the History of the Seventh Corps Area and Charles Winter's Nebraska Territory Postal History, which coincidentally has a neat little postmark from the Saratoga Post Office. But on initial survey, nothing that indicates an established fort in Saratoga.
Are you following the Company or more interested in the geography/location in Saratoga? If your focus is on Saratoga, I suspect you're aware of Ham, Eggs, and Corn Cake - might there be anything of use in Beadle's diary? (Love your work, full disclosure.) If you're tracking the company, Civil War Archive has pretty solid regiment-level troop movement based heavily on Dyer's Compendium. And, of course, the state and county historical societies may have additional material.
While 1861 falls outside the scope of the Historic World-Herald Archive, Chronicling America does have a small handful of Nebraska papers that reach back to the era, and they definitely buzz with war news. You might consult the Bellevue Gazette, the Nebraska Advertiser, and the Woodsman's Echo and test out different searches.
This is 100% speculation but it seems most likely that troops would have been quartered in the abandoned Saratoga Springs Hotel.
This is the sort of question that will haunt me for a little bit, so if I stumble across any interesting sources I'll follow up. Our local history research staff may also be able to recommend other sources for you if you email firstname.lastname@example.org.