What would be expected of a member of a militiaman in medieval Scandinavia in terms of arming himself, and how would he go about it?

For Sweden.

From the earliest surviving written Swedish county laws (and probably for quite some time before that) peasants were by law required to own and train with arms and armour, in lieu of taxes. The older Västgöta law was probably first compiled and written down around 1220, although earlier examples may have existed, and the oldest surviving copy is from around 1281. In theory these requirements ended with the Alnsö law from 1280, which created the Swedish nobility and gave tax exemption for anyone who could raise an armoured rider (ie a knight). The county laws varied on what equipment peasants were required to bring to the ledung (the peasant muster fleet), and rarely got into detail. It is still debated on wether the laws represented an ideal rarely reached on the ground or if they requirements were realistic and often met. It is also debated on how long the tradition of ledung is - some argue that it goes back to at least 1034, when Ingvar Vittfarne travellered east with a host to aid relatives among the Kievan Rus against the Pechenegs - the rune stones erected in remembrance of those that died on that campaign after it reached the Caspian sea are so evenly distributed that some say that it had to be a ledung, while others don't think the evidence is conclusive as no runestones mention a ledung and the organisation of the land into the hundare used by the ledung seem to have happened in the late 1100s or early 1200s.

That said, for armour, the peasant militia often used captured armour from previous battles - as is evident by the drawings by the German landsknecht engineer Paul Dolnstein of Swedish peasant militia he fought as part of Danish King Hans' campaign to regain the Swedish throne in the (pardon my language) clusterfuck that was the latter 2/3 of the Kalmar Union. The peasants are fighting in organised ranks, have unit symbols, banner, use locally made swordspears and are armoured in plate or coat of plate cuirasses and use helmets that were in vogue about 100 years earlier.

The peasant militia buried at the mass grave at Visby after the Battle of Korsbetningen 1361 were still dressed in their torso armour, but had their weapons, shields and helmets removed when looted. That said, most arms and armour seem to have been made locally, by local smiths, if one is to believe the armour found in the mass graves. They vary a LOT in age, design and even type. Most are coat of plates of varying design, but a lamellar armour (probably at least 200 years old when used at Visby) was there as was chainmail. Most of the men seem to have worn coifs and probably had iron hat helmets in top of that (that were removed and looted before striking many of the by that time wounded peasants lying on the ground in the head to kill them).

Medieval Swedish peasants travelled to things to see law metered out, disputes settled, announcements made and elect Kings (at least before 1544) and trade. Arms and armour could be sold there, either transported by traders or made locally. Surplus captured arms could also be available here. In times of unrest or revolt, many would seek to arm and armour themselves, so travelling merchants would go where the demand was.

What would be accessible would vary a lot. Swedish peasants inherited armour from their fathers and augmented it by looting defeated mercenaries if they could. In the Nordic 7 years war, the Swedish King at the time considered buying armour and arms from the peasants to raise new troops, but found that the peasants after 40 years of peace had not maintained their arms and armour, and what was available were mostly outdated by two or even three generations - and this was in the 1560s.

Sweden has always been rich in iron, and iron was relatively cheap. That does not mean that arms and armour was not a major investment and prohibitively expensive. Most Swedish peasants owned their own land and were represented at the things and later the estates parliament - combined with their arms and armour, this gave them economical, political and military power, and they guarded this power jelously, rising many times to protect it from foreign and internal threats to reverse their 'old rights'. They could thus afford arms and armour, even if rarely top of the line and if the mass graves at Visby and Dolnstein's drawins are to be believed, rarely arm and leg armour.

I am not aware of any attempts to restrict arms and armour ownership from the peasants - considering the fiery way the peasants defended their economical and political influence, one can probably assume they would fight just as fiercly against any attempts to encroach on their military power. The limiting factor was cost rather than law.

So, to summarise. A peasant would inherit arms, grow cash crops to sell at the market to buy new ones to hire the local smith to make him some. If he has some means, he could either travel or send a message to a dedicated armour or arms smith (which were few and far between on the countryside) to make him higher quality things. One or two generations old armour would probably be available, either to buy from neighbours or on the market at the thing. Daggers, swords, spears, two-handed spears, swordspears, axes, two-handed axes, glaives, hunting bows and hunting crossbows seem to have been common arms for the Swedish peasants. Two-handed spiked clubs (see the center weapon) were also common, as were conversions of tresher flails into weapons. The only thing really limiting a peasant is his economic and social resources to buy arms and armour, and where he can travel to acquire them.