The usual method was to obtain iron/steel and/or ready-made weapons by trade or as tribute. Weapons would tend to come from civilised neighbours (for the Eurasian steppe, major sources of weapons were China and the cities of Central Asia (including Iran). Iron could come from the same civilised neighbours, or forest tribes to the north of the steppe.
If iron/steel rather than ready-made weapons, then a blacksmith would turn it into armour and weapons. Being nomadic isn't a big problem for a smith - a basic kit for forging is simple and not too heavy, and forging doesn't need as much fuel as smelting, nor fuel of as good quality (dried dung will suffice).
For more on obtaining metal and nomadic blacksmithing, see https://www.reddit.com/r/AskHistorians/comments/hryymk/metalworking_among_the_nomadic_steppe_tribes/
A small portable forge isn't the best thing for trying make large pieces of plate armour; this is one reason why much armour made on the steppe was lamellar, using small plates like:
A complete lamellar armour might look like:
Steppe helmets were often laced-together lamellar helmets:
The relatively small smelting furnaces used by the forest peoples to the north of the steppe are fine for producing iron, but due to their small size are usually poor for making high-carbon steel. A portable steppe forge is also not the best thing for uniformly heating a sword blade to the ideal temperature for quench-hardening, so a sword made on the steppe from northern forest iron is unlikely to be a sword of very good quality. The best swords are likely to come from the southern civilised neighbours of the steppe. At times, such weapons would only reach the steppe by smuggling, due to selling arms to the steppe peoples being forbidden.
Two European visitors to Mongolia reported on Mongol military equipment. John of Plano Carpini, who travelled to Mongolia and back in 1245-1247, wrote:
Moreover they are required to have these weapons: two long bows or one good one at least, three quivers full of arrows, and one axe, and ropes to draw engines of war. But the richer have single-edged swords, with sharp points, and somewhat crooked. They have also armed horses, with their shoulders and breasts protected; they have helmets and coats of mail. Some of them have jackets for their horses, made of leather artificially doubled or trebled, shaped upon their bodies. The upper part of their helmet is of iron or steel, but that part which circles about the neck and the throat is of leather. Some of them have all their armour of iron made in the following manner: They beat out many thin plates a finger broad, and a hand long, and making in every one of them eight little holes, they lace through three strong and straight leather thongs. So they join the plates one to another, as it were, ascending by degrees. Then they tie the plates to the thongs, with other small and slender thongs, drawn through the holes, and in the upper part, on each side, they fasten one small doubled thong, that the plates may firmly be knit together. These they make, as well for their horses as for the armour of their men; and they scour them so bright that a man may hold his face in them. Some of them upon the neck of their lance have a hook, with which they attempt to pull men out of their saddles. The heads of their arrows are exceedingly sharp, cutting both ways like a two-edged sword, and they always carry a file in their quivers to sharpen their arrowheads.
The points to note are (a) the description of lamellar armour, (b) swords are only owned by the richer people, while axes are used by others, and (c) horse armour (of rawhide) is used.
Marco Polo, more concisely, wrote:
their arms are bows and arrows, sword and mace; but above all the bow, for they are capital archers, indeed the best that are known
which substitutes a mace for the axe. Notably, a wrought iron mace is a perfectly acceptable mace - hardened steel, desirable for the edge of a sword, is unnecessary. While a steppe-made sword might be inferior, a steppe-made mace will work quite well.