In this case, "mount" = animal you ride on. Most likely a horse. To be dismounted = to not be riding a horse; or, to get off your horse (either voluntarily or involuntarily) -- the use of passive voice here strongly hints at the "involuntary" scenario. "fetch" = "bring".
Thus, "I can bring your horse if you got off/fell off/were knocked off the horse during a battle."
That said, "own" doesn't really belong here. I'd have to see the full context to understand what it's doing.
@Quinn1 Well, it's not entirely clear to me. When he says "your own mount", it implies that the horse you had before wasn't your own, so in that case it would indeed mean a new horse. Or, it may mean that if you lost your horse, he will go looking for your horse specifically.
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