I used to be sure that I knew the answer to that question. I thought it was the Kearny Scots, the earliest mention of whom that I've seen is in 1895. Then I learned that the Kearny Scots who were among the founding members of the National Association Foot Ball League in 1895 were not really the same team as the Kearny Scots who won five straight American Soccer League titles around 1940 and who still exist today as an amateur team. That original Scots team folded in 1919, and the current one was founded in 1931.
Of course, 1931 is quite a long while ago, too, enough to make the Scots still a good contender for that "oldest still-operating" title, but I know of four older ones. The oldest of those is the Beadling Soccer Club of Bridgeville, Pa., a suburb of Pittsburgh. The Beadling club was founded in 1898 and played for decades in excellent Pittsburgh-area leagues. It has always been an amateur club, and it won the National Amateur Cup in 1954, beating Simpkins of St.Lous in the final. It also has done well in the U.S. Open Cup, reaching the quarterfinals four times and the semifinals once. Today, it operates a series of youth teams in various age groups along with its senior team.
The other three are Hoboken FC of Hoboken, N.J., founded in 1912; Gremio Lusitano of Ludlow, Mass., founded in 1922, and Croatian Eagles of Milwaukee, founded in 1922 (I didn't know about Hoboken or Croatian Eagles before Big Soccer poster Chris Edgemon wrote about them last year).
Of course, there were many soccer teams in America before 1898, hundreds of them, but none of them have come close to lasting until 2012 that I know of. Many of the earliest, going back as far as 1866, were organized just for the day or for a few games. The first one that seems to have been organized on something of a continuing basis was at Princeton College, which played its first game in 1867 against the Princeton Theological Seminary and then played at least one game in each of the following nine years. There were association football teams in the late 1860s and early 1870s at several colleges that play soccer today, but those teams are not continuous. Those colleges, such as Columbia, Princeton, Yale, Rutgers and a few others, mostly abandoned soccer in 1876 and didn't resume playing it until decades later (the oldest of the revived teams in Haverford's, founded in 1905).
After the colleges faded out of the picture, the first two club teams to be organized on a continuing basis were the East Ends of Fall River, Mass., and the Hibernians of St. Louis, and which one of them came before the other is not completely clear, at least not to me. The East Ends apparently were formed in 1880 by members of the Fall River East End Cricket Club, but there are no records of games played by the East Ends before April 1883. Meanwhile, the Hibernians were founded in 1881, and the first recorded game played by them is in November 1881. Both the East Ends and the Hibernians were disbanded in 1891.
I haven't mentioned the Oneida Football Club of Boston, who played from 1862 to 1864 and have sometimes been called the first soccer team in America. Research in recent years has indicated that the game they were playing, called the Boston Rules Games, was so much like rugby that it's incorrect to consider it an early form of soccer. Nevertheless, the Oneidas were the first team in the United States in any form of football to be organized on a continuing basis.
But back to the original question. For now, I think that the oldest still-operating soccer club in the United States is Beadling. However, considering that I once before thought that I knew the answer to that question and later found that I was wrong, I'm open to suggestion. If there is another club out there that beats Beadling, it would be by only a few years, however.