'Request' is Latin for 'seek again' — from 'quaerere' (to seek). The polite end of the asking family.
An act of asking politely or formally for something; to ask for something politely or formally.
From Old French 'requeste' (a request, demand), from Vulgar Latin *requaesta, from Latin 'requīsīta' (things sought for), feminine past participle of 'requīrere' (to seek again, to ask for), from 're-' (again, back) and 'quaerere' (to seek, to ask, to inquire). The PIE root is *kʷeh₂s- (to seek, to ask). The sense development runs: to seek again → to ask for → a polite asking. 'Require' is a doublet, entering from the verb directly; 'request' arrived via the noun form. 'Query,' 'quest,' 'inquest,' and 'question' all share
English has both 'request' and 'require' from the same Latin source 'requīrere.' The split happened because each word entered English through a different route: 'request' came through Old French 'requeste' (a noun that softened to a polite asking), while 'require' came more directly from Latin 'requīrere' and kept the stronger sense of demanding something as necessary. Same root, different registers.