Riding High promises to let you do just that. Except for the third-person cover-based part. You won't be shooting anything here; you'll just be owning a virtual horse.
While the official site is a little bare, the game's Facebook page has recently posted answers to a number of questions, confirming, for example, that the game will feature distinct personalities for each horse and that players will be able to engage in PvP activities. (Probably not horse fights.) If this really is all you ever wanted from online gaming, you can register for the future beta test on the official site.
[Thanks to Zenaphex for the tip!]