In various English dialects the verb to dap means ‘to move quickly and lightly; to trip along’:
North Yorkshire: He goes dapping along, as if he were on springs.
Somerset: I was always quick to dap about on my feet.
Devon: Her were terribly spry. Her dapped round like anything. — Joseph Wright, ed., English Dialect Dictionary, vol. 2, D–G, 1900.
The number of times a stone skips across water does so in a series of daps, and The Welshman, 9 Feb. 1855, describes waterfowl as “dippers, dappers, divers”: a dipper like a duck or swan, a diver like a cormorant, and a dapper like a kingfisher that skims the surface of the water.
According to Wright, the verb was common across Ireland, Yorkshire, Wiltshire, Gloucester-shire, Wales, the West Country, and the Isle of Wight.
Transferred to a vertical plane, it means ’to bounce’, as in the description of the first football play in Cardiff (1873–4):
There were twenty players a-side, the ball was round, not oval, and was never to be picked up off the ground, but had to be dapping, if only a few inches. — Evening Express, Final Football Edition (Cardiff), 7 Dec. 1907.
I’m trying to envision football with basketball rules but it just wouldn’t be the same.
Dapping is also a special fishing technique where the bait or fly on a very short line is bobbed rapidly in the water. This usage has also reached America:
One of the major advantages of dapping is that you avoid entering the stream at all. Too often, wading into a creek either disturbs the trout or puts them down completely. — Field & Stream 91/12 (Apr. 1987), 116.
It makes sense, then, that a shoe designed for “moving quickly and lightly” would eventually recruit this word in at least some of the dialects where it was found.
Although some sources, such as the Concise New Partridge Dictionary of Slang (2nd ed., 2015), date the print debut of daps, ‘ rubber-soled canvas shoes’, to 1924, daps were tripping lightly across Cardiff somewhat earlier, at least when there was no mud:
This must have been a great disappointment to the men wearing “rubber daps,” as with these it was impossible to get a foothold, and it was anticipated that fast time under these conditions would be out of the question. — Evening Express, Football Edition (Cardiff), 12 Feb. 1910.
The quotation marks suggest a fairly new and/or slang word, so calling these shoes daps in print likely means that in 1910, people in Cardiff hadn’t been doing so for all that long in everyday speech. The shoes occasionally make appearances in Cardiff and Bristol newspapers thereafter:
…and Manley, who, taking with him four pairs of leather soled daps, eight pairs socks, and a pair of leather shoes, arrived in Berkeley last evening, … — Western Daily Press (Bristol), 8 Sept. 1922. BNA (paywall)
…but we are urgently in need of daps, shorts, and light sweaters, so that the boys may get the full advantage of the training … — Western Daily Press (Bristol), 8 Mar. 1923. BNA (paywall)
She even said there was no need for little Betty to take her daps out on the pavement and shake the sand out of them. — Western Mail (Cardiff), 30 Aug. 1923. BNA (paywall)
My search of relevant newspapers yielded no earlier usage of dappers with the same meaning as daps, though an ordinary web search has ample results today. This suggests that daps is not an abbreviated form of dappers, but that the longer word is a more modern extension.
This does not hold, however, for the original verb, which may very well be a derivative of dapper.
Dapper entered English from Middle Dutch/Low German dapper, ‘bold, strong, sturdy’ (Cf. ModGer tapfer, ‘brave, courageous’) in the mid-15th c. with the familiar meaning of ‘elegant, neat trim’, but somewhat later expanded to include ‘quick, nimble’ and by the 17th c. ‘small and active, nimble, brisk, lively’. You get to brisk and lively from heavy, strong, and sturdy the same way Robin Hood called his companion “Little John.”
…with her a little lad naked and blind, yet did i note, that bow and shafts he had, and two wings to his shoulders fixt, which stood like little Sayles, with farre more various colours mixt, then be your peacocks Tayles; i seeing this little dapper Elfe, such Armes as these to beare, quoth i thus softly to my selfe, what strange thing haue we here, i neuer saw the like thought i: t’is more then strange to me, to haue a child haue wings to fly, and yet want eyes to see; … Michael Drayton, The Muses Elizium, 1630. EEBO
Since this “dapper elf” blind Cupid is naked, it’s not likely Drayton has a snappy dresser in mind, but certainly something nimble and quick.
A derivation of dap from dapper in the quick and nimble sense does equal service, however, for those who’d prefer to be snappy dressers, which may have contributed to extending daps to dappers after the verb became obsolete.
One folk etymology that doesn’t work, however, is deriving daps as an acronym from the factory name “Dunlop Athletic Plimsoles,” as Dunlop did not take over the original manufacturer, the Liverpool Rubber Company, until 1925, fifteen years after the first mention in Cardiff.