To my knowledge, there is no direct surviving evidence for leather armor in the Viking Age. The texts don't mention it (except for a reference to magical reindeer hide in one of the sagas, but that's clearly not a normal thing), and nothing from an archaeological context suggests leather armor was worn. We have a similar lack of evidence for leather armor across all of early medieval Europe - there's almost no evidence for its use.

There are a few ways to interpret this. The safest would be to say that leather armor didn't exist, but we have to be cautious here. Our archaeological evidence isn't a random crosssection of society - most of what we have is from graves and settlement sites. Armor of any kind is rarely found in graves, and while this might mean that armor was rare among the living, it's more likely to indicate that armor was usually inherited instead of being buried with the dead, except in extraordinary cases. At settlement sites where people live, we usually only find trash and ritual deposits - again, armor doesn't show up. But that might just mean it's not being thrown away or given as a butied sacrifice. Add to this the fact that leather - and iron too, for that matter - needs special conditions to survive in the ground without decomposing, and you have a good recipe flr archaeological invisibility. So absence of evidence here really might not be evidence of the absence of leather armor.

When you consider how people fought, though, the possible lack of body armor might make more sense. Battles were fought in a shieldwall, where every fighter was covered by his/her own shield and the shields of his/her neighbors, and the formation would ideally only break apart after the enemy was running away. A warrior's shield was his armor.

That leaves the head exposed still, so you would assume helmets to be common; but here we have a lack of evidence (only one helmet has been found in Viking Age Scandinavia - so we're back to the problem of absent evidence).

Iron and leather were both expensive - you have to kill a 3 year old cow for thick leather, and that leather has a lot of other possible uses (including providing the rawhide with which shields were covered); so it may well have been the case that leather was more useful making shields and other necessary items than it would have been for body armor. And iron ore took an incredible amount of charcoal to smelt (turn into useable metal), meaning that every man may not have been able to afford a metal helmet. The lack of evidence makes informed speculation difficult, but it's certainly possible that your ordinary Viking Age fighter may not have been able to afford the materials for a helmet or other armor, leather or otherwise.

The other alternative is cloth armor. Padded armor is surprisingly effective at stopping weapons on its own (it works a lot like kevlar, with multiple layers of fabric slowing down and often stopping a cut), and it was worn in Byzantium shortly before the beginning of the Viking Age. There's no evidence, to my knowledge, of its use in Scandinavia, but it seems just as likely as leather armor for body protection.

This is all a very long way of saying, we really don't know. I'd personally suspect that most Vikings wore a few layers of thick tunics at a minimum, because even a few layers of fabric might slow a poorly placed strike. But it's quite possible that any form of real armor was limited to the military elite, those who could afford the investment in their expensive materials, with ordinary fighters trusting in the shieldwall to keep them alive.