From Old English 'āscian,' PIE *h₂eys- (to seek) — dialectal 'ax' is not a corruption but a 1,200-year-old pronunciation predating 'ask.'
To say something in order to obtain an answer or information; to request something from someone.
From Old English 'āscian' (also 'āxian,' 'āhsian') meaning 'to ask, to inquire, to demand, to call, to summon,' from Proto-Germanic *aiskōną (to ask, to demand), from PIE root *h₂eys- (to wish, to desire, to seek). The Old English form had several dialectal variants — 'āscian' in West Saxon, 'āxian' in Anglian — and the modern pronunciation with /sk/ rather than /ks/ reflects the West Saxon form. The dialectal form 'ax' (from Anglian 'āxian'), often stigmatized today, is actually the older and more conservative pronunciation, attested continuously since the eighth century. Key
The pronunciation 'ax' (as in 'Let me ax you a question') is not slang, bad English, or a modern corruption — it is the direct descendant of the Anglian Old English form 'āxian' and was standard English for centuries. Chaucer wrote 'axe' in the Canterbury Tales. The form 'ask' won out because it came from the West Saxon dialect that became the written standard, but 'ax' has been in continuous use for over 1,200 years.