Horse & Pony Books in the Beany Malone Series:
The Malone series is a longish girl's/family series which is for the most part non-pony although does contain a couple of horse-themed books which I include below.

See the full series here

READING ORDER:

1) MEET THE MALONES
(CROWELL [USA] 1943)
Reprinted a number of times in hardback and paperback.
Most recently reprinted in paperback by Image Cascade.
SUMMARY: Book 1 in the series. The Malone children are allowed to go their own way so no-one is that surprised when Beany's elder sister Mary turns up with an injured black horse she has bought instead of a new dress! But when their grandmother arrives to stay, things begin to change as she tries to make the children comply with her own ideas of how they should behave. Mary realises she may have to take responsibility for her younger siblings.

2) BEANY AND THE BECKONING ROAD
(CROWELL [USA] 1952)
Reprinted a number of times in hardback and paperback.
Most recently reprinted in paperback by Image Cascade.
SUMMARY: Book 4 in he series. Beany takes a road trip with her brother and a horse named Quaker.

Collector's Info:
These have been reprinted many times and are very easy to find in the USA, although not so easy elsewhere.

Other Horse & Pony Books:

THE GYPSY BRIDLE
LITTLE, BROWN & CO [USA] 1930)
ILLUSTRATED BY KURT WIESE
Available as a print on demand edition.
Original edition rare.

RIDING HIGH
(CROWELL [USA] 1946)
ILLUSTRATED BY GERTRUDE HOWE
Reprinted in hardback by Crowell.
SUMMARY: Horsy teenage romance. Emily and her horse Pal O'Mine are friends with Kip and his horse Pinto, but gradually romance develops.

Collector's Info:
Gypsy Bridle is rare and can be quite expensive, although it is now available as a print on demand edition. Riding High is easier to find but still not overly common. Both are very hard to find outside the USA.