The short answer: Yes, it is possible, but the different parts of the process are usually divided among specialists.

It is possible for the whole process:

• collect iron ore,

• make charcoal,

• smelt ore to make iron,

• work the bloom from the smelter to expel slag and produce fairly pure iron,

• forge the iron to make tools/swords/armour

to be done by a single smith (or smith and an assistant or a few assistants). For example, this has been the case in some times and places (e.g., West Africa, as described by Kense (1983)).

However, it is often more efficient for many of these tasks to be carried out by different people. For example, if the iron ore is bog iron, it can be collected by farmers during slack times of the season. Other tasks are best done by people with specialised skills (underground mining of ore, charcoal making, smelting, forging), and dividing the work between different specialists means that no one individual needs to master all of the skills (they only need their particular subset). Specialisation can also allow large-scale production at some levels (typically mining, charcoal making, and smelting), with improved efficiency. Even within tasks such as making a sword, separate parts of the job such as forging of the blade, the grinding and polishing of the blade, making the hilt, and making the scabbard can be split among different specialists.

If a single smith does make the whole sword, it often has a relatively simple hilt, with a simple iron guard and pommel. Some examples of Chinese "village jian" (and dao), usually made by a single smith, are available in https://forum.grtc.org/viewtopic.php?t=893&p=6995

Don't expect a single smith to make fancy plate armour! First, there are many specialised skills involved, and second, mechanisation such as water-powered hammers - something not readily available to a single smith - is very useful. See comments by u/WARitter on specialisation in armour-making in https://www.reddit.com/r/AskHistorians/comments/4nod2n/did_medieval_blacksmiths_tend_to_specialise_is/ (water-powered grinding wheels are also useful in blade-making, and can usefully result in separation of grinding blades from forging and hilting). Armours made with small pieces of iron are more easily made by a single smith, so body armours such as scale, lamellar, or jacks (e.g., see https://commons.wikimedia.org/wiki/File:A_member_of_the_Tudor_Group_makes_a_Jack_of_Plates.jpg for a modern replica being made), and helmets made of pieces riveted together (and lamellar helmets, where the pieces are laced together) are much more likely than plate. However, even armours such as these benefit from specialisation.

Reference:

F. J. Kense, Traditional African iron working, University of Calgary, 1983.