During WWII, it would have been the Quartermaster Corps that handled vehicle contracts. The contract was awarded to the "parent" company who was to make the product - called the Prime Contractor. The Prime contractor was then able to sub-contract work to other companies as needed due in part to speed up production. You might have a sub-contractor who made the rear axle assembly, the transmission, the brake assemblies, wiring, etc.. Some military items that would be specific to the contract were often supplied by the government and called GFE or Government Furnished Equipment.
Some Prime contractors had little use for sub-contractors as they produced all parts themselves in house (like Ford), while other Prime contractors sub-contracted the majority of their work out and did the final assembly. All military goods, trucks, planes, machine guns, etc. had to be inspected by a military inspector (who was on-site) and then "accepted" the item/assembly and credited it towards meeting the total number of items as agreed to between the Prime contractor and the government. An initial order could have been for 500 X-items and if the order was to be extended for more items, the government would renegotiate for the additional order/price and write an "addendum" to the original contract order, ie 500 X-items and "Addendum 1" for 450 X-items more. The government could also cancel a contract at any time.
The Prime contractor was paid a partial payment up front to provide cash flow so the Prime contractor could begin setting up for production. Some would secure war loans through the banks to get going quicker and then use the first government installment to pay towards the loan. The government could divide the remaining monies owed the Prime contractor by production items "accepted" or one larger payment when the contract was completed. The Prime contractor was responsible for negotiating pricing & paying any sub-contractors.
Trucks were not let out to the public during the war years. I think sales/manufacturing stopped by March 1942. In 1944, an allotment of trucks were approved by the government to be sold to the public to replace worn equipment such as farm equipment or for industrial uses. I have a booklet, which I can't seem to place, that gives the total number of trucks released to the public. I think it was 10,000 of all makes. This is why you don't see many trucks having war year production dates unless they were earlier production vehicles held in storage or those released in 1944. My brother has a neighbor who's father was a farmer and he got a 1944 production (VIN year) Dodge chassis that has a stake bed on it. I think it is a 1 1/2 ton truck with dual tires (not twin axles). 1944 VIN dated trucks are rare.
The quality of the truck during the war versus after the war should not have changed much unless some items were substituted with an equivalent that was not as durable, but worked, due to war material shortages.
Here is a great WWII war movie about International Harvester detailing what IH made during the war. "The Strong Shall Be Free."
https://www.youtube.com/watch?v=3j14hLO6Pw8