Yes, there were, starting in at least the 14th century -- the records we have attest to, for example, Lord Howe having a staff of five trumpeters and seven drummers in 1470, as part of a general retinue of a great man at sea. By 1513, the King's council specifically appointed three trumpeters to "attend upon" the Lord Admiral, Edward Howard, as part of the creation of that office. By 1582, we know that each of the Queen's ships, down to pinnaces, were allotted at least a drum and fife, and in the larger ships there was a "noise" of three or four trumpeters (this is possibly my favorite collective noun, by the way). From N.A.M. Rodger's The Safeguard of the Sea, chapter 22 in my electronic edition:

[the trumpeter] should have a silver trumpet, and himself and his noise to have banners of silk of the admiral's colours. His place is to keep the poop, to attend the general's going ashore and coming aboard, and all other strangers or boats, and to sound as an entertainment to them, as also when they hail a ship, or when they charge, board or enter her…

(Rodger is there quoting Sir William Monson in his Tracts.)

In 1595, during Drake's and Hawkin's voyage, they brought along "sundry instruments of music" for eight musicians (not specified) and nine trumpeters, who would also have other duties aboard ship -- during Drake's circumnavigation, the trumpeter and the chaplain were the watch-standers aboard Golden Hind when the Marigold sank in a storm off Cape Horn.

This was not limited to strictly Royal Navy ships (and in this period the line between royal ships and private ships acting in the monarch's interest was blurry anyhow). The Levant Company's ship Centurion was attacked by a Spanish ship in the Gut of Gibraltar in 1591, the English seamen used music to rally themselves. Again from Rodger:

there was a sore and deadly fight on both sides, in which the trumpet of the Centurion sounded forth the deadly points of war, and encouraged them to fight manfully against their adversaries. On the contrary part, there was no warlike music in the Spanish galleys, but only their whistles of silver, which they sounded forth to their own contentment.

In a time well before flag-signaling, the trumpeters also played a role in conveying orders or information to other ships -- in 1594, Robert Dudley ordered his trumpeters to contact an unknown ship:

hailing them with his noise of trumpets made them know their duty unto our English colours by vailing [furling] their topsails.

(Dudley's Voyage, George Warner.)

By the 1620s, the trumpeter's position had risen to that equivalent of a warrant officer -- by 1627, one watch-bill says the trumpeter should supervise the handling of the mizen-sails, which is sensible given their usual station on the poop.

I hope this is helpful -- let me know if there is more that I can answer.