Some dictionaries list horsy as the primary spelling of the word that can be (1) an adjective meaning horse-like, (2) an adjective meaning of or having to do with horses, or (3) a diminutive of horse. But these dictionaries are behind the times. Horsy had a brief heyday in the middle of the 20th century, but horsey was unquestionably preferred before 1940 and is again preferred today. This is the case throughout the English-speaking world.
There is also horsie, which is used most often as the diminutive of horse, but it is rare compared to the other two.
In Google News searches covering a large selection of major English-language publications and limited to 2000 to the present, most instances of horsy are in the New York Times, which seems to be an outlier on this issue. But outside the Times (where horsey does also appear, but not as often as horsy), horsey is the far more common spelling. Here are few examples from around the English-speaking world:
They stock the barn with all kinds of horsey goodies: brushes, blankets, water buckets, feeders, saddles, stirrups, bridles, bits, reins. [Denver Post]
Slightly older but less weathered by an outdoor horsey lifestyle, Ann has certainly been busier since the nuptials were scheduled. [Sunday Express (U.K.) (dead link)]
Horsey movies are usually buddy flicks, partnering the steed with often young and diminutive pals. [Globe and Mail (Canada)]
Those behind the scheme are convinced that the polo lifestyle as much as the horsey discipline will swing it their way. [New Zealand Herald]
It requires impeccable timing and a straight horsey face. [Manila Standard]
This ngram graphs the occurrence of horsie, horsey, and horsy in American books, magazines, and journals published between 1800 and 2019:
And this ngram shows the same in British texts:
Both graphs show a fleeting rise in the use of horsy in the middle of the 20th century, but it’s only a brief blip in the otherwise long, unquestionable ascendancy of horsey. These graphs might be skewed a little by Horsey being a surname (we see no instances of horsy as a surname—though it probably exists), but that has nothing to do with the rise and fall of horsy.