In Scandinavia? Anyone. Regional laws varied but as far as i'm aware no area forbade or even got involved with some sort of weapon control. The viking era norsemen did not write down their laws, the had lawmen who memorized all the laws. The earliest written one I can think of right now is Gutalagen which was the law for the Swedish island of Gotland. It was written in 1200s, so later than you wanted and more christian influence but it doesn't mention weapons other than a guard is to have weapons after he turns 20 years old - so perhaps age was a factor in who could own a sword or not, an 18 year old guard might not of been considered mature enough to bear arms, especially if he had to use them wisely.

Now, if we look at archeological evidence and try to determine who owned what from that; we know that even women were sometimes buried with axes, swords and even shields. At Flisa in Norway they've even found a woman buried with all three. (The reason why these weapons were buried with the women is unclear, but that they owned and used them is a pretty popular belief). In Birka there are also women who were buried with spears and swords and one young girl with a dagger. Now, these women weren't necessarily all warriors or anything but they could certainly use what they were buried with. Women were certainly in a good position in Scandinavia compared to the rest of the world but they were still not always at the top of the social hierarchy, so if women could own swords then practically anyone could ( other than slaves ).