At the end of WWII the Allied powers ended up with huge numbers of captured Wehrmacht weapons. They collected these from captured or killed troops during fighting on the Eastern and Western fronts, or from overrunning territory (including supply depots) on those fronts, or simply from the process of Germany's surrender and subsequent occupation by Soviet, French, British, and American forces.

Almost all armed forces during the mid-20th century had official plans and procedures for making use of captured arms, some more than others. In the case of ex-Wehrmacht weapons there were huge numbers of them (in some cases hundreds of thousands to millions of particular models) and many of them were still regarded as decent quality weapons in the immediate post-war era, so they were kept around and put to use. In particular the StG-44 was still near the top of its class in the immediate post-war era, and the MP-40 was a well liked high quality sub-machine gun that used incredibly popular and widely available ammunition.

There are basically two ways that such weapons made their way to Vietnam and into the hands of the Viet Cong. The first is that both the MP-40 and StG-44 were issued to French colonial forces in "French Indochina" in the post-war era. These could then fall into the hands of the Viet Cong as individual units were killed or captured or as weapons depots were overrun and captured.

The second is that the Viet Cong could acquire them via logistical supply from the Soviets et al. Particularly in the immediate post-WWII timeframe the small arms shipped to the Viet Cong by the Soviets included a mix of Soviet weapons along with a lot of ex-Wehrmacht surplus. This included not only MP-40s and StG-44s but also Pak-40 light artillery, Mauser 98k bolt action rifles, machine guns like MG-34s, and even FG-42 paratrooper rifles. Over the course of time these supply shipments shifted towards more Soviet and Eastern bloc made small arms such as SKSs and AK-47s. Additionally, many of the German weapons, particularly weapons like the StG-44, began to be used less as the WWII-era ammunition stockpiles began to dry up.

Edit: I should also note that both the MP-40 and StG-44 (as well as many other ex-Wehrmacht weapons) ended up in the hands of all sorts of forces all over the world, sometimes due to the same sort of logistical support by the Soviets, sometimes through more convoluted routes. The weapons were in use in the Korean war, the "Malayan emergency", and many other conflicts. StG-44s (now basically collector's items) have been spotted in all sorts of seemingly unlikely places post-WWII, from Algeria and Somalia in the '70s to Syria in the 2000s.