Webster gets it wrong again. The (Scottish
) Gaelic pònaidh
and the Irish (Gaelic) pónaí
are just borrowings from the English.
The English word is of Scots origin (that is, Scots English not Gaelic), and only became common around 1800. The first recorded use is powny in 1659, in a Scottish text. In 1710 we read of "Union Ponies, a Kind of Horses foaled upon the Borders, and occasionally owning either Country"; and in 1730 we get a gloss "Pony, a little Scotch horse". Fergusson wrote powney in 1774 and Burns wrote of pownies in 1785.
The origin is not known for certain but the suggestion is that it is from an earlier form *poulney, from Old French poulenet, diminutive of poulain 'foal', from Latin pullus 'young animal'. To prove this we need documentary evidence of an earlier Scottish or Northern word in pol-, poul-.
So much for fact. There is an oft-quoted folk etymology (a kind of urban legend) that the word is related to Epona, a Gaulish horse-goddess, whose worship was widespread among Continental Celts. (Her Welsh equivalent was Rhiannon, and her Irish Macha. The Mabinogion tells how Rhiannon was forced to carry visitors to court as punishment for alleged infanticide.)
This is just sheer coincidence. Someone has seen "Epona", seen "horse", and thought, "oh that looks a bit like 'pony'", and concocted a fantasy. Gaulish was a P-Celtic language, like Welsh, and Welsh has ebol 'colt, filly', which I'm guessing is related. Irish and Scots Gaelic are Q-Celtic and the corresponding word would contain EQ- or EC- (and in fact appears as ech-). You can't get the Scottish word powny from a Celtic ep-.
In fact this latter stem, to veer off on what is now an irrelevancy, is the common Indo-European for horse, *ek'wos, which gives us Latin equus and Greek hippos and Persian asb and Sanskrit as'va and Old English éoh (as in the Tolkienian horse-lord names Éomer and Éowyn). But that's not important right now.
Interestingly, the Welsh for 'pony' is not a borrowing from English, but merlyn. Isn't that charming? I don't know where it comes from, but I suspect march 'horse' + glyn 'glen'. But don't take my word for it, that could be an entirely spurious guess. It does not, of course, have any connexion with the wizard, alas. The Welsh name for Merlin is Myrddin. The change to an -L- possibly came about as the romances spread to France, because of the titters among the fine ladies in their wimples and liripipes at mention of the great, mysterious, and powerful wizard Shittin (you know about merde, of course).
Of the slang senses: '25 pounds' is recorded from 1797, 'crib' from 1827 (U.S. origin), and pony up 'settle up, pay up' is recorded from 1824, also U.S. origin. None of these early quotations in the OED makes it at all clear how the slang sense arose.