I know Marvel's Thor uses an axe (Jarnbjorn) and sword (Odinsword) in the comics, but an acquaintance of mine insists that the Thor of the norse Mythology also used them.
I couldn't find any source that explicitly mentions Thor using such weapons.
Is there a source claiming Thor used an axe and/or sword in the Norse Mythology?
No, those weapons are inventions of the comics.
We do have a couple of stories in which Thor can not use Mjölnir. In Þrymskviða, the hammer has been stolen by the giant Thrym who demands the hand of Freya in order to give it back. Thor is dressed up in bridal clothing to retrieve his hammer, so it is difficult to say whether the apparent lack of other weapons is part of his disguise or due to lack of options.
In the story of Þórsdrápa, also told, with variantions, by Snorri in Skáldskaparmál, Loki tricks Thor into going on a journey to defeat the giant Geirröðr without the help of Mjölnir or his belt of strength. Apparently, he has no alternatives, because we are told that Thor meets the friendly giantess Grid, who gives him good advice and lends him her belt of strength, a pair of iron gloves, and a staff called Gríðarvölr. He uses both the staff and the gloves to defeat enemies he meets.
So, no, Thor does, afaik, not use either sword nor axe.
