To my knowledge, there is nothing in the Slayer oath that specifies weapons.
" I am a Dwarf! My honour is my life and without it I am nothing. I shall become a Slayer. I shall seek redemption in the eyes of my ancestors. I shall become as death to my enemies, until I face he that takes my life and my shame. "
—Gotrek Gurnisson, swearing upon the Oath of the Slayer.[2a] Slayers | Warhammer Wiki | Fandom
Is from a Wiki, says nothing about weapons. In lore you find that the Greataxe with 2 blades is the Runesymbol of Grimnir and as such the favoured weapon of Slayers. But a Slayer may use nearly anything to kill their enemies.
Following Quote is from the Story: Slayer´s Honour by Nathan Long Gotrek and Felix - The Anthology:
" „Agnar pushed off the wall and stepped again in front of Gotrek, blocking the way to the balcony. ‘Why did you attack the orcs just now, when there was a troll before you?’ He asked. ‘A true slayer should attack the most dangerous foe.’
‘I killed the orcs to distract the troll with their meat,’ said Gotrek, with surprising restraint. ‘It made it easier to kill. Now let me by.’
‘Easier to kill?’ Agnar shook with rage. ‘Easier to kill? A slayer does not make his enemies easier to kill!’
‘For Sigmar’s sake, lower your voice!’ said Felix. Nobody paid him any attention.
‘Does he not?’ asked Gotrek. ‘Why do you carry that axe?’ Agnar blinked, confused. ‘If you wanted to make your enemies harder to kill,’ said Gotrek. ‘You would attack them unarmed, yet you don’t.’ ‘
An axe is a Slayer’s weapon!’ said Agnar. ‘It is tradition. That’s not the same as–’
‘ Grimnir asks of us that we fight our enemies with all our skill and strength, ’ said Gotrek. ‘ Anything less is suicide, which he disdains. Do you think he means us not to use our strength of mind? I fight with all the strength I possess.’“"
Which for me is a sensible approach to being a Slayer, since being a Slayer is not simply seeking suicide to redeem yourself, something that many people seem to forget.
Apart from that, I think the Slayer itself is not well executed by GW, since they need massive amounts of Plotarmor to survive any amount of time, but well that falls under personal opinion.