An early version of 'blow your own horn' appears in Sir Thomas Chaloner's 1549 translation of Erasmus's 1509 (first printed in 1511) Encomium Moriae, "In Praise of Folly":
And what (I praie you) maie be more apt or better fittyng, than dame Foly to praise hir selfe, and be hir owne trumpet? for who can livelier descrive me, then I my selfe?
The use in Erasmus is figurative. 'Trumpet' appears at least as early as 1447 in the sense of a "means or agent (real or imaginary) which proclaims, celebrates, or gives warning of something" (OED, trumpet, n., sense 3).
The Latin source of Chaloner's translation is
Quid enim magis quadrat, quam ut ipsa Moria, suarum laudum fit buccinatrix....
From that source ("buccinatrix"), it is apparent that the intended sense of "trumpet" in Chaloner's translation is the figurative sense of 'trumpeter' (OED, sense 4b of trumpet, n.) corresponding to OED's sense 3 of 'trumpet' (given above).
In light of the much earlier appearance of versions of 'blow your own horn' in the UK than in the US, it is more than likely that the US versions derive from UK versions.