The surface meaning in many spirituals is indeed based on chariots in the Bible, whether carrying people to heaven or to victory.
However, often that sense only served to conceal a deeper meaning about escaping from slavery, and this meaning itself had two levels. One level was as a general metaphor for the longing to escape, to be carried away from a life of suffering, whether by fantasy or by death. The other sense was as a specific secret code referring to the Underground Railroad.
'Underground' in this sense isn't literally 'unterirdisch,' like a mine or a subway, but figurative, like 'Untergrund-,' meaning secret, undercover, in hiding. It was not a real railroad but a network of people who helped slaves travel to the North and then to Canada to reach freedom. Safe houses were called 'stations,' white organizers were called 'conductors,' and black travelers were called 'cargo.' So in that sense, a chariot could have been a coded term for a wagon, maybe as a pun on a railway car as part of the insider lingo, or as a reference to any vehicle used to transport slaves secretly.
If I understand the German correctly, I'm not sure 'Versorgungswagen der Yankee-Truppen' is quite right either. The Underground Railroad was already in existence several decades before the Civil War. And the Civil War was not fought in the first instance to free the slaves (though many people who sympathized with the North hoped that that would also be a result), but to prevent the southern states from seceding. Lincoln's Emancipation Proclamation didn't come until 1863, halfway through the war.
Selkie or someone may know more about all this, but my own impression is that especially in the earlier years of the war, there wasn't necessarily even any organized plan for what to do with freed slaves in areas occupied by northern troops. The Union Army was mainly busy just fighting the war, so the slaves may often have had to fend for themselves even after the armies passed through.