What a fascinating question!
I hate to report something negative, but I spent a fair bit of time last night researching this and had no great luck.
I checked a couple of letters of Pliny and Marcus Aurelius that I thought could have mentioned women riders but didn’t find anything. Pliny does have one ambiguous letter (To Fuscus, 108) about what he did at his villa, and he mentions eating with his wife and riding but it is not clear if she went with him during his rides.
I found a couple things that may be of interest.
1) Tullia used a chariot to trample her father’s body (Livy 1.48). She didn’t travel on horseback according to the myth.
2) Good old lecherous Ovid mentions trying to seduce travelling women several times (Art of Love 1.487 and Remedies for Love 663), but they are only mentioned with litters.
3) The Oppian Law, which forbade some luxuries to women during the financial crisis of the Punic War states that women can’t “ride in a carriage drawn by horses in a city, or any town, or any place nearer than one mile (Livy 34.1).” Riding on horseback is not mentioned. Livy places an interesting speech in the mouth of Lucius Valerius where the alternative of women riding in a carriage was “to follow on foot (Livy 34.7).” It is as if riding on horseback was not even considered.
4) Plato called for the training of women in the art of horsemanship (Laws 804e+). This was not Rome, of course, but the implication is that women in the Greek world (or perhaps only Athens?) did not ride. Rome was deeply infused with Greek influences, even in archaic times.
I did find
one episode of a female rider in the Roman world. This is the Jesus nativity story, of course. Mary’s ride is most explicit in a non-canonical gospel.
Quote: And he saddled the ass, and set her upon it; and his son led it, and Joseph followed… And they came into the middle of the road, and Mary said to him: Take me down from off the ass, for that which is in me presses to come forth. And he took her down from off the ass, and said to her: Whither shall I lead thee, and cover thy disgrace? for the place is desert.
Infancy Gospel of James 17.