OED does provide what, absent another offer, must be considered the most comprehensive historical account of the development of the phrase 'to speak for' through various nuances of meaning...up to and beyond the sense of "taken" (equivalent to 'engaged' used in the sense of "secure[d] (something) for one's own or another's use or possession").
Sense 1 of 'to speak for' is
To make a speech or plea in place of or on behalf of (a person); in later use esp. to plead for. Also, to make representations concerning (a thing). speak for yourself: expressing a desire to dissociate oneself from what another has just said or the assumptions behind it.
OED being a historical dictionary, sense 1 is sense 1 because it is the earliest attested sense of the phrase the OED scholars found. It is first attested sometime before (a) 1300:
a 1300 K. Horn 171 Hor[n] spak for hem alle.
Observing the "in later use esp. to plead for" nuance noted in the definition of sense 1, a later attestation evidences that development:
1535 Bible (Coverdale) 2 Kings iv. 13 Hast thou eny matter to be spoken for to the kynge?
That "later use" itself exposes the source of the development of the nuance captured in sense 2:
To beg or request; to ask for.
This nuance is attested first with
1560 Bible (Geneva) Song of Sol. viii. 8 What shal we do for our sister..when she shalbe spoken for?
Note the sequence of dates. Sense 1 is first attested a 1300; one nuance of that sense, attested 1535, develops into sense 2, attested 1560.
Sense 2, then, develops further, as attested by two additional quotes for sense 2:
1594 J. Lyly Mother Bombie i. iii. sig. B3, They giue vs pap with a spoon before we can speak, and when wee speake for that wee loue, pap with a hatchet.
1608 Shakespeare King Lear iv. 240 The shame it selfe doth speake for instant remedie.
Through use and repetition, sense 2 develops another distinct nuance, represented in the definition of sense 3:
To order; to bespeak; to engage.
The development of this sense is also plainly traced in the attestations provided:
Ordered, in the mercantile sense:
a1688 J. Bunyan Israel's Hope Encouraged in Wks. (1855) I. 583 As your great traders do with the goods that their chapmen have either bought or spoke for.
1730 N. Bailey et al. Dictionarium Britannicum at Bespeak, To speak for something; to give order for it to be made.
1816 J. Austen Emma I. xv. 273 The bell was rung, and the carriages spoken for.
1859 H. B. Stowe Minister's Wooing xii. 115 Three months beforehand, all her days and nights are spoken for.
Engaged [= Taken]:
1943 Sun (Baltimore) 25 Feb. 6/1 (advt.) We hope to preserve even more food this year. But well over half of this season's pack is already spoken for by the Government.
1971 Petticoat 17 July 29/2 He's not married, but he's involved, as they say, spoken for, and has lived with his girlfriend in London for the last few years.
From sense 3, the more abstract sense 4, "To indicate; to betoken", first attested in 1832, develops. Sense 5 is a parallel line of development for 'to speak for yourself' into the more abstract "to speak for itself, to be significant or self-evident", first attested 1779.
The OED's ancestry story of the phrasal verb ('to speak for') is not the complete or final word. Hairs may be split over the nuances of meaning representing the development of the phrase; such arguments would have to be supported with a detailed analysis of the context of the attestations given, and might also involve other attestations not shown in the OED story. Nonetheless, without significant work essentially retracing work already undertaken and completed by previous scholars, the OED's story is likely to be the best on offer.