This comes up constantly and the answer does not change: horseback riding is not permitted on the beaches of Hilton Head Island. There is no operator running beach rides here, no seasonal window when it becomes allowed, and no way to arrange it privately.
The nearest place it genuinely happens is Daufuskie Island, across Calibogue Sound, where trail operators run rides that include the island's beach. Daufuskie has no bridge. You get there by boat, and the practical route for visitors is the public passenger ferry from Broad Creek Marina on Hilton Head to Freeport Marina on Daufuskie, a crossing of roughly 45 minutes. Once there, transport is golf carts rather than cars.
Plan it as a day, not an outing. Ferry schedules are limited and seasonal, rides need booking ahead and separately from the ferry, and missing the last boat back is a real problem rather than an inconvenience. Confirm current ferry times and ride availability directly with both operators before you commit, because both change with the season.
Inland, there are riding barns around Bluffton and the wider Lowcountry offering lessons, boarding and arena work rather than trail rides for casual visitors. If your family rides seriously and wants more than a walking trail ride, that is where to look, and it is worth calling ahead to ask what they offer visitors.