North America did, in fact, have horses (actually, the genus Equus evolved in North America), which went extinct 11,000-13,000 years ago:
Though some survived by crossing the Bering Strait land bridge into Eurasia, evolving into the European horses of today.
Of course, if you were assuming that no one ever brought any animals over to NA, and that horses stayed extinct, and the selection pool was only native wildlife, you'd still have moose, oxen, the occasional deer/elk, reindeer, bison, etc.
Though deer or an elk may not be able to support much weight, North America still has oxen and buffalo. Oxen have been used as transport and for farming, and originated in the Americas and Europe; if all else fails, dogsleds have been in use up north for thousands of years. And there is still moose, which can survive even a direct impact from a car and if you managed to domesticate them would make pretty decent mounts; running at 35 mph and swimming at 6 mph (though science says that moose and other large animals make pretty poor mounts, if only due to their feeding habits).
For transport, oxen, buffalo, dogs, and even reindeer work and have worked just fine; if you need a war mount, again, charge a moose into battle and strike fear into your enemies' hearts as the almighty creature stomps them into the ground.
Point being, North America does have a lot of potential mount animals, in fact, horses (the most popular of the bunch) originated from the Americas, so with time and a lot of work put into domestication, a stone-age America would have mounts just as viable as any other continent's.
*Note: I'm aware that moose and buffalo can't be ridden! I've seen a bunch of articles about it; however, I did assume that OP wanted just a list of anything remotely mount-worthy and wouldn't need super in-depth science about it. Apologies for any misconceptions!