First, iron tools were important for many purposes: woodworking, farming, cooking, and more.

Second, iron ore is common, and iron can easily be made locally. Often, the limiting resource is fuel, rather than ore. Large deposits of iron ore suitable for mining might not be available locally, but bog iron was usually available - bog iron was the most commonly used iron ore in the early Medieval period in northern Europe (and in many other parts of Europe, and other places in the world, and other times, too). Bog iron has advantages: collecting bog iron is easier than hard-rock mining, and the ore is relatively pure (iron sand shares these advantages, but is less widely available). The disadvantage is that it is more sparsely spread, and is uneconomical to use for modern industrial-scale iron-making (which was not an issue in the Viking Age). Given wood for making charcoal, it is usually possible to make iron locally. Thus, the local demand for iron tools can be met locally. Most iron would have been locally-produced.

A smith who makes iron tools can also make weapons. The most most common iron weapons were usually locally-made: spears, arrowheads, axes, and knives. It was common for spearheads to be all-iron (rather than steel, or iron with steel edges). While this is an inferior method of construction, it is easy and cheap, and does the job. Iron spears continued to be common in later medieval Europe, and many halberds and similar polearms were also all-iron. All-iron spears were also usual in much of sub-Saharan Africa into modern times, and these were most certainly functional weapons. Knives and axes are much more likely to have steel cutting edges.

Swords, on the other hand, were typically much fancier weapons, and were much rarer. Some swords appear to have been locally-made, by non-specialists (i.e., by a blacksmith rather than a specialised swordsmith). Many sword blades and swords were imported. In the Viking Age, the most important sword-making region appears to have been the Rhineland, in Frankia. The Carolingian Empire often banned the export of weapons (at least to their enemies), but Frankish swords still made their way into Viking hands. Frankish blades appear in Scandinavia, in Britain, and in Ireland with local styles of guards and pommels, so it's likely that many of these blades were exported as unmounted bare blades, and hilts were made locally by the buyers. The classic "Viking sword" is perhaps better called the "Carolingian sword" or "Viking Age sword" since they're found well outside the Viking area (e.g., southern Europe).

Many of the good-quality sword blades show decorative pattern-welding with a high standard of workmanship. Many have distinctive inscriptions or inlays which have been interpreted as maker's marks or branding (and counterfeit branding in some cases). One interesting group of swords, the Ulfberht swords, named after their VLFBERHT inlays, consists of 4 different types of swords. One group are all-steel, which is very unusual for European swords of the time. The high carbon content led to the tentative identification of their steel as crucible steel, probably imported. Analysis of the composition of these blades shows that a likely origin is Khorasan, which is quite plausible, since (a) the region (especially the Ferghana Valley) was a major exporter of crucible steel, and (b) trade routes from there went along the Volga and to the eastern Baltic. Having passed through the hands of Viking traders, perhaps this steel was made into swords by Viking smiths. The alternative is that the swords were made by Frankish smiths.

The next two groups of Ulfberht swords are pattern-welded. One group is hardened, and the other is not. At least the hardened group is usually assumed to be Frankish-made - these are typically excellent blades, with harder steel edges than the crucible steel Ulfberhts. The last group is all-iron (and therefore unhardened), with highly variable inscriptions - these are often assumed to be low-quality forgeries.

Wherever they were made, the pattern-welded ones were made by specialist smiths. Other weapons are found with pattern-welded blades, including knives and spears. Some of the pattern-welded spears were probably made from broken sword blades (this appears in literature), while others were made as fancy spears in the first place.

Al least some iron armour was imported from Frankia - mail and helmets.

Not all weapons were iron. Shields were mostly wood (with iron bosses). These were made from split planks, glued together, and possibly/probably sometimes/often faced with linen or hide. Bows were wooden self-bows - the classic longbow. Spear hafts were usually ash, using either split timber or coppiced poles. Finally, that very important weapon for going viking: the warship. These were made locally, using split timber.