Yes, infantry platoons had their own carts - typically a small wagon with one or two small horses ('Panje' horses) in 1942-1945 on the Eastern front.

An infantry division was (April 1940) supposed to have 16,860 men, 1,743 riding horses (especially for officers, scouts and couriers), 3,632 cart horses and 895 horse carts. The quantity of cart horses and carts had to be increased after immense horse losses in late 1941; the replacements were more resilient, but also weaker Russian horses, and their smaller carts.
Often times former Red Army soldiers (especially minorities) were employed as volunteers for many logistical tasks - charioteer was a typical task. These volunteers often received German uniforms, but without rank insignia. Some even received iron crosses and other medals. Speaking about such volunteers in German language is of course not very PC.


Typical rifle company:
3 horse carts each 2 horses or 3 horse carts each 1 horse + 1 cart with 4 horses
1 field kitchen cart with 4 horses
1 supply train (Verpflegungstroß I) with 1 horse cart + 2 horses

Companies were also supposed to have some 3 metric ton trucks. Military vehicles were required to have a marching gear for 4 km/h (foot march speed). Civilian vehicles were more typical post-mobilisation, of course.

Quick source: "Das Handbuch der deutschen Infanterie", Alex Buchner,