This is an interesting question. During the SGW arc books, I don't recall much reference to any animals being used in warfare at all, though I will see, as I am currently rereading them.
So, we have several things to consider. First of all, the United States could very well be quite motorized. They have the large auto factories of the North, Especially in Michigan, Indiana, and Illinois. THey also have all the resources they need to take advantage of that, such as the steel producing regions in Pennsylvania and West Virginia, and the oil of California, Dakota, and Canada. THe books also give me no indication that the US is any less motorized than OTL, especially given that in WWII, the motorization of the US Army was in large part due to the industry of the North, and the large amount of draftees from urban background who possessed familiarity with automobiles, or complete mechanic skills. THe US Army would definitely take advantage of this against the Confederacy, especially in the western theaters where they would need to make use of it to pin down the Confederates in the open spaces west of the Mississippi while most of the manpower and effort is going towards the east.
I do think you might see more use of animal power in some secondary theaters. An example being the campaign in Utah, where the Mormons have no real motorization of their own, nad some of the fighting is in mountainous terrain. Pack mules might also see a lot of use in the Appalachians, where the terrain is quite similar to that of central Italy, where the US Army used them as well.
On the COnfederate side, we might look at Turtledove's basic inspiration, the Wehrmacht. The Confederacy lacks the same scale of industry the Germans had, and the Germans themselves were always short of trucks. Another factor to consider is that the South of TL-191 is also more motorized than the South of OTL, after Featherston pushed ahead the program of mechanization in order to render negro labor irrelevant and clear the way for his Population Reductions. The Confederates would probably, like the Wehrmacht, concentrate as much of their available motor vehicles with their armored units for the same reason. You might also, in the later stages of the war and out west, see a lot of Confederate "technicals", civilian vehicles confiscated and quickly given welded on weapons. But overall, the Confederates would probably see a lot more usage of animals, especially for use in the Appalachians, as I mentioned before, and for rear area tasks to free up trucks for the front.
One final thing that goes unmentioned in the SGW books, but which I think bears mentioning, is cavalry. In spite of the machine gun, cavalry enjoyed a last hurrah in WWII on the Eastern Front, where the large open spaces, rugged terrain, and presence of several less motorized nations contributed to their usage. The Soviets had several dozen cavalry divisions, mostly Cossacks, and the Germans, Hungarians, and Romanians all used cavalry. Given the advent of lighter, more portable machine guns, infantry antitank weapons, and light artillery in the form of mortars, I might suggest that the Western Theater of the SGW would see a brief revival in cavalry fighting as mounted infantry in the open space of Texas, Sequoyah, Kansas, COnfederate Mexico, Missouri, Arkansas, and so on. Possible even further east, in terrain such as the swampy areas of the Deep South (The Russians used large numbers of cavalry in the Pripyet Marshes in Belorussia) and so on.