How many trucks did Russia and Germany have at the start of Barbarossa?

Soviet Union: On 22nd June 1941, the Red Army had around 270,000 trucks, and received another 745,000 during the war. Out of these, 150,000 were new domestic production, 221,500 trucks drafted from the industry and agriculture sectors, 60,600 captured enemy's trucks and 312,600 lend-lease trucks.

The average lift capacity of a Soviet truck was 2 tonnes, and they had very poor cross-country capability (two wheel drive), except for LL trucks which were mostly 2.5 tonne capacity with 6x4 drive.

The Soviets used 600,000 horses in 6/41, in 1-8 horse teams, less 60k for riding. Soviet divisions almost never had the authorized levels of transport, and were usually at 25-40% of the authorized level.

Source: varies, but mostly from Operation Barbaross Vol IIIA by Askey.

In the WWII, the basis of logistic (Wehrmacht and Red Army) was a horse. The infantry company (Wehrmacht / RKKA) had one wagon with one horse for supplying ammunition. At a higher level of supply (battalion, regiment, division), the Wehrmacht and the Red Army used trucks. During major offensives, the supply route stretched out up to 100 kilometers. However, for example, the Finnish army did not have many trucks. Many Finnish divisions were supplies only by horses.

In addition, the Wehrmacht at the beginning of the war had a strong transport aviation fleet. This allowed the supply of advancing units or surrounded garrisons.

The next level of supply is the railways. These are not only large echelons, but also small trains (the Wehrmacht often used captured armored cars with one railway carriage). By the way, the Finnish army had in 1939 a rocked railway to supply the Mannerheim Line.

Unlike Germany, the USSR did not have a developed industry. In fact, in the early 20s there was only 1 plant - AMO (and Yaroslavl workshops). In the 30s, Ford built a factory in the USSR. But it produced light 1.5-ton trucks GAZ-AA (FORD-AA). It was the cheapest and most produced truck. ZIS (formerly AMO) plant built the best and perfected Soviet ZIS-5 trucks. The Yaroslavl plant built small series of heavy 5 ton trucks. But they were complex, expensive. The USSR was able to create many light 1,5 and 3 ton trucks, but did not have heavy trucks and all-wheel drive trucks.

In Germany there was a developed industry. There were many firms that in the 1920s already produced 3 ton trucks. Wehrmacht had excellent heavy trucks in service. For example, the "backbone of supply" in France were 10-ton trucks. Fortunately, the USSR did not have German autobahns, thus these trucks could not be used in the USSR. It's Russia, baby!

Fuel. The Wehrmacht used excellent synthetic gasoline. The USSR had very poor gasoline for trucks. The main truck of the Wehrmacht (Opel) is simply stalled after Russian gasoline. In fact, in addition to "General Frost" and "General Dirt", the Wehrmacht fought also with "Colonel Chemical". The Opel company was forced to "modernize" the Opel to the level of the Soviet ZIS-5: simplified the design, forced the engine to eat a surrogate.

In the period from 1928 to 1937, the number of vehicles in the Red Army increased by 1050 units to 40 thousand cars. By the beginning of the Great Patriotic War in the Red Army was in service with 272.6 th.(thousand) cars, including 257.8 th. trucks and special [trucks], of which the overwhelming majority of cars wore GAZ-AA and ZIS-5 brands. [1, p.108]

As of June 22, 1941, there were 272.6 thousand cars of various types and 5784 armored cars in the Red Army. Until the end of the year, another 166,300 vehicles were mobilized ... [1, p.366]

Wehrmacht had half a million cars. And these were good trucks, including off-road trucks. In 1941, 333 thousand cars were produced in Germany, 268 thousand in the occupied countries, and the satellites of the Third Reich produced 75 thousand more cars.

On June 22, 1941, the share of 1.5ton's trucks accounted for 151.1 thousand units, 3-ton - 104.2 thousand units, that is almost 94% of the entire park. On January 1, 1942, taking into account the mobilization of machines from the national economy and their losses were 148.7 and 89.1 thousand respectively. [1, p.368]

...from June 22, 1941 to May 9, 1945, all Soviet plants produced 205 thousand cars of domestic design, of which 146.6 thousand were sent to the Red Army. machines, or 71.5% of their output. At the same time, according to various sources, from 312.6 to 477.8 thousand cars arrived in the Lend-Lease in the USSR, that is, 2,1-3,2 times more than put all their Soviet factories [...] In addition, the Red Army had 60.6 thousand captured vehicles at the end of the war. [1, p.371]

40% - the all-wheel drive cars. On May 1, 1945, the entire of the Armed Forces of the USSR amounted to 664.5 thousand cars, of which 100th. units were in the automobile troops, numbering by that time 35 automobile regiments, 173 separate battalions and 30 companies.

Sources:

1) Evgeny Kochnev, "Automobiles of the Red Army 1918-1945" / Moscow, "Yauza", "Exmo", 2009

To answer your side questions about paucity of information about Soviet industry: having established a superb intelligence network in the West, USSR was understandably paranoid about the symmetric efforts and classified all information about its industry to the degree that no official numbers could ever be trusted. Combine this with приписки and you see that no one, not even Politburo, knew what was actually happening. Compound that with the obsession to downplay the impact of lend-lease and you will see why we will never know anything.

Another issue one has to be aware of is quality vs. quantity. The Lend-Lease trucks and jeeps were far better than anything the Soviet industry could produce at the time. The "workhorse" was "полуторка" - 1.5t truck based on Ford Model AA, and the Soviet Army had about 150,000 of them (and their modifications) in 1941.

When accounting for the German trucks using production numbers, one has to be aware that in 1941 the whole Europe was working for them: they were getting Renault, Citroën, and Tatras in addition to their own production.

For the Operation Barbarossa the Germans deployed 104 infantry, 19 panzer and 15 motorized infantry divisions (plus some other unidentified divisions). Nominally, this amounts to 104,100 trucks, 96,200 motorcycles and 67,240 cars. Wikipedia claims about 600,000 motor vehicles (about twice as much), which sounds reasonable - a lot of transport has to be done by non-front-line units at the corps/army level.