Yes they did!
By the fourth century A.D., the production of arms, armor, and military equipment was centralized into state-owned factories/armories known as fabricae. The origins of the system is unclear, though it is likely their codification occurred during the reigns of either Diocletian or Constantine. Precisely how the system was administered is also unclear, though the Notitia Dignitatum suggests it was part of the portfolio of the Magister Officiorum, a senior court official.
Fabricae were dispersed throughout the provinces, and were responsible for producing the vast majority of the army’s equipment. The relatively simple construction of Late Roman ridge helmets hints at the ability of the fabricae to mass produce military equipment, so that the Empire certainly retained the ability to provide armor to the bulk of it’s soldiers. Fabricae were located near the frontiers, yet were constructed far enough inside the interior to protect against raids. They were also constructed along major roadways to facilitate transportation, and it is likely they existed inside urban circuit walls, though their exact placement is unclear as archaeologists have yet to positively identify the remains of a fabrica.
Along the Rhine and Danube frontiers, each frontier province had a fabrica scutaria, and each frontier diocese had two fabricae which produced more general equipment. Other fabricae specialized in the production of certain types of equipment, such as artillery, cataphract armor, bows, and swords. It is also likely that each fabrica, regardless of specialization, also produced more general equipment in small numbers. The production of military clothing was also relatively centralized, though production appears not to have taken place in fabricae, but rather in other state-owned facilities located near major fabric producing centers, like Alexandria, though army-administered weaving mills were located throughout the Empire, such as at Reims, Tournai, and Trier. The production of military clothing was paid for by a special tax aptly known as the vestris militaris.
Although the production of military equipment was relatively centralized by Late Antiquity, Roman soldiers certainly would not have presented a uniform appearance. The goal of the fabricae was not to ensure blanket uniformity, but rather to centralize the production of military equipment under Imperial control, and to ensure quality of production. All Roman units would have had a “local flair”, and appearance, while likely being relatively uniform within units, would have varied across units. Certain basic equipment, like javelins and arrows, were likely produced by regimental armorers. Clothing would also have been replaced by local wares, as cotton and linen clothing tends to deteriorate relatively quickly in the field.
The other response does a great job with weapons factories, so I thought I would discuss the grain dole: The problem tracing an individual corn from crop to belly is that state did not rely on any single source for its provision. Rome was a city of a million people, and while the state apparatus that supported that was indeed impressive that does not mean it wasn't always to a point a few steps away from catastrophe. For example, the emperor Claudius was an effective administrator who presided over a period of peace and prosperity, but in the winter of 51 CE a poor Nile flood was still enough to provoke famine in Rome--although it is worth noting that Suetonius says the crowd pelted him with bread crust, so it doesn't seem to have been quite starvation-famine. So even in the best of times a supply shock could cause real hardship.
Therefore, the supply of grain was a mix of private and public, and the sources were a mix of private and public. There was no government owned merchant fleet, but some merchant ships, particularly those coming from Alexandria, were privately contracted out by the government in order to transport grain. These ships made up some of the largest vessels seen until modern times, the Isis as described by Lucian being fully 55 meters in length. Other ships coming in to port were privately owned and operating on their own initiative, attracted either by the ready market in Rome or for the privileges that came with carrying grain (such as the ability to bypass tariff duties). In all, the state probably leveled enough in-kind taxation in Sicily, North Africa and Egypt to feed the city if nothing went wrong, but merchants on their own initiative provided usually insurance against supply shocks in any grain producing province.
As for the source, this was again a mixture of private and public sources. As mentioned, in certain provinces the Empire levied its taxes in grain, and so this can be seen as a major source of supply. There would also be those either from cash tax paying provinces or simply merchants on their own initiative who would sell grain to the grain fleet. But a major source were the imperial estates, which constituted large swaths of North Africa and Egypt. These estates, however, do not seem to have been directly managed by the imperial bureaucracy, and were instead leased out to tenet farmers or renters. So all in all there is a similarly complex ecosystem of suppliers.
A good source is Erdkamp's The Roman Grain Market.