An interpretative sign at the Mumma Farm in Antietam National Battlefield in Sharpsburg, MD reads in part, “After the war, the Federal Government compensated residents for damage caused by Union soldiers. However, since this farm was burned by Confederates, the Mummas received no compensation.”
There was no agency in charge. The Constitution allowed people to petition the government for redress: so Samuel Mumma petitioned Congress for relief. This was a very common practice- a great deal of the House of Representatives' time in the 18th-19th c. was taken up by people asking for the government to pay them for damages from wars ( for example, if Federal troops took all your chickens without paying) or for services rendered to the country- Revolutionary War soldiers had to petition for a pension and have their pensions approved on case-by-case basis.
These requests/petitions, called Memorials, were often prepared with the help of lawyer who would assemble supporting documents , then presented by the petitioner's congressman, then referred to a committee for consideration. If the committee approved it, thought it valid, it often was approved if there were no objections.. ( sometimes there were. The congressman of the family of James Rumsey petitioned for something in recognition of his building a steamboat, but it was contested by the congressman of the family of his rival John Fitch, who'd built another steamboat- and it was turned down) . After the House approved, like any bill it had to be approved by the Senate before the pension , etc. was paid out. You would think there would have been a revolt by the congress for having to deal with so many small claims, but there wasn't, and an amazing number of memorials were processed. You can see Samuel Mumma's petition here was presented as H.R 4477 ( Dec.1 1873-June 23 1874) by his Maryland Congressman, Lloyd Lowndes,Jr., and then referred to the Claims Committee. Which is where someone scrutinizing the evidence likely found fault. Notice that there are quite a number of claims for relief, for compensation, from others in the South.
Most of these small matters eventually would be handled by the civil service: after the Civil War congress created an agency to handle veterans' benefits, for example. But at this point in time, there was no such agency to deal with Mumma..