The phrase one’s trumpeter is dead is applied either to a habitual boaster or to a person boasting on a particular occasion. (Here, the word trumpeter denotes one who extols.)

The image is that, when one’s trumpeter is dead, one is forced to find one’s own trumpet—as explained by the English antiquary and lexicographer Francis Grose (1731-1791) in A Classical Dictionary of the Vulgar Tongue (London: Printed for S. Hooper, 1788):

Trumpet. To sound one’s own trumpet; to praise one’s self.
Trumpeter. The King of Spain’s trumpeter; a braying ass. His trumpeter is dead, he is therefore forced to sound his own trumpet.

Incidentally, therefore, the phrase the King of Spain’s trumpeter designated a braying ass ¹—as also indicated in the following definitions from the same edition of A Classical Dictionary of the Vulgar Tongue:

Donkey, Donkey Dick. A he, or jack ass: called donkey, perhaps, from the Spanish or don-like gravity of that animal, intitled also the king of Spain’s trumpeter.
[…]
Spanish, or King of Spain’s, Trumpeter. An ass when braying.

¹ According to A Dictionary of Catch Phrases British and American, from the Sixteenth Century to the Present Day (Taylor & Francis, 2005), by Eric Partridge and Paul Beale, King of Spain’s trumpeter, in the sense of a braying ass, is

a neat pun on the pun: Don Key—donkey.

These are the corresponding definitions from the first edition of A Classical Dictionary of the Vulgar Tongue (London: Printed for S. Hooper, 1785)—in which there was no entry for the word trumpet:

Donkey, donkey dick, a he, or jack ass, called donkey, perhaps from the Spanish, or don like gravity of that animal, entitled also the king of Spain’s trumpeter.
[…]
Spanish, or king of Spain’s trumpeter, an ass when braying.
[…]
Trumpeter, the King of Spain’s trumpeter, a braying ass; to sound one’s own trumpet, to praise one’s self.

The first known user of the phrase one’s trumpeter is dead was the statesman, inventor and scientist Benjamin Franklin (1706-1790) in the first letter that he wrote under the pen name of The Busy Body to Andrew Bradford (1686-1742), the publisher of The American Weekly Mercury (Philadelphia, Pennsylvania). In this letter, published on 4^th February 1729, Benjamin Franklin explains that he is proposing, with Andrew Bradford’s allowance, “to make Use of the Weekly Mercury as a Vehicle in which [his] Remonstrances shall be convey’d to the World”:

As I do not aim at publick Praise I design to remain concealed; and there are such Numbers of our Family and Relations at this Time in the Country, that tho’ I’ve sign’d my Name at full Length, I am not under the least Apprehension of being distinguish’d and discover’d by it. My Character indeed I would favour you with, but that I am cautious of praising my Self, lest I should be told my Trumpeter’s dead: And I cannot find in my Heart, at present, to say any Thing to my own Disadvantage.

Benjamin Franklin used the phrase again in a letter to the clergyman, physician and agronomist Jared Eliot (1685-1763), written from Philadelphia on 13^th February 1749—as published in The American Journal of Science, and Arts (New Haven: Published by S. Converse, for the Editor, 1822):

What you mention concerning the love of praise is indeed very true, a love of praise, although corrected by art reigns more or less in every heart; though we are generally hypocrites, in that respect, and pretend to disregard praise; and that our nice modest ears are offended, forsooth, with what one of the ancients calls the sweetest kind of musick. This hypocrisy, is only a sacrifice to the pride of others, or to their envy; both which I think, ought rather to be mortified. The same sacrifice we make, when we forbear to praise ourselves, which naturally we are all inclined to; and I suppose it was formerly the fashion, or Virgil, that courtly writer, would not have put a speech into the mouth of his hero, which now-a-days we should esteem so great an indecency, Sum pius Æneas,—fama super æthera notus ². One of the Romans, I forget who, justified speaking in his own praise, by saying, every freeman had a right to speak what he thought of himself as well as of others. That this is a natural inclination, appears, in that all children show it, and say freely, I am a good boy; am I not a good girl? and the like; ’till they have been frequently chid, and told their trumpeter is dead; and that ’tis unbecoming to sound their own praise, &c.

² Sum pius Æneas,—fama super æthera notus translates as I am pious Aeneas,—known by fame beyond the heavens. It is a quotation from the Aeneid, a Latin epic poem by the Roman poet Virgil (Publius Vergilius Maro – 70–19 BC) which relates the travels and experiences of Aeneas after the fall of Troy.

One Ephraim Doolittle used the phrase in a letter published in The Farmer’s Library. Or, Vermont Political & Historical Register (Rutland, Vermont) of 15^th April 1793—in this letter, Ephraim Doolittle appealed to the public in his own vindication, because someone was blackening his character:

I am not conscious to myself, that I ever wittingly or willingly injured any man to the value of one copper; but perhaps my trumpeter is dead, or only sick.