This got too long for a comment, so I hope it's all right to post here. The Writing in Margins blog is mine, so I'd like to offer a few addendums.
It's always hard to disprove something rather than prove it, but I really haven't found any books of Welsh folklore that mention corgis as fairy steeds. There's Giraldus Cambrensis' Itinerary of Archbishop Baldwin Through Wales (1191) where fairies ride tiny horses and have tiny hunting dogs: "They had horses and greyhounds adapted to their size." There's no indication that the fairies ride the dogs. If they swapped dogs with horses, he would probably mention that.
Fairy corgis don't appear in Thomas Keightley's Fairy Mythology, John Rhys's Celtic Folklore: Welsh And Manx, or any of Katharine Briggs' books that I know of. Rhys's Volume 1 does have a line "Like Welsh fairies, the Manx ones had, as the reader will have seen, horses to ride; they had also dogs, just as the Welsh ones had."
In general, European fairies are seen riding horses. If the fairies are miniature, then the horses are miniature (as in A Pleasant Treatise of Witches, 1673).
It hapned one day, as this admirable Semstress sate working in her
Garden, that casting aside her Eye on some fair Flower or Tree, she
saw, as she thought, a little Gentleman, yet one that shew'd great
Nobility by his cloathing, come riding towards her from behind a bed
of Flowers; thus surprised how any body should come into her Garden,
but much more, at the stature of the person, who as he was on
Horseback exceeded not a foots length in height; she had reason to
suspect that her eyes deceived her. But the Gallant spurring his
Horse up the Garden, made it not long, though his Horse was little,
before he came to her:
There are exceptions. In Walter Map's 12th century work "De nugis curialium", King Herla meets a dwarf who rides on a goat. And in the British fairytale of Tom Thumb, first recorded in the 17th century, Tom Thumb (who has a fairy godmother and fairy clothes) rides a mouse. However, I don't know of any cases where dogs are the steeds. I would love to hear if someone else knows of one.
For me, the timing says a lot. Corgis have been around for a long time, but they were not formally recognized as a breed until the early 1900s, and it also took time to differentiate the two types, Pembroke and Cardigan. Wikipedia's saying they weren't recognized as distinct until 1934. Anne Biddlecombe was one of the first famous Pembroke Corgi breeders and would have had a strong interest in laying out precisely how Pembrokes were different from Cardigans. The "fairy saddle" is a distinct Pembroke marking. The poem specifies that it's about the "Pembrokeshire Welsh Corgi."