The answer is that some horses can initiate a grapple, strictly speaking. The only means a horse has of obtaining a grip would be it's bite attack, since hooves are not exactly known for their ability to grip things. Only war horses have bite attacks. I suppose a non-war horse could use their mouth, but they would not be proficient.
Even then, however, a natural horse would be unlikely to initiate a grapple. Simply put, it's not how they fight. You couldn't convince a horse to grapple any more than you could convince a horse to use a dagger +5 in it's mouth. Horses are prey animals. They tend to fight to get out of a grapple rather than to start one, and they panic when they are not standing. This is why even injured horses are kept standing (also because it causes intestinal problems to lie down as a horse -- they lack the connective tissue that would keep their guts from tying in knots). As a DM I would only allow horses to grapple in defense, and they would always try to escape.
In the case of a paladin mount, they would possess the intelligence to understand what the PC wishes, but, still, this is not how a horse fights and the player's influence on their mount is not absolute. It's not dominate monster. The player would know the mount would be hesitant to even begin to do such a thing. It would be much happier if it could kick, since that is it's primary natural weapon. As a DM, I would allow the mount to grapple offensively, but the mount would really not be happy about it. If it happened repeatedly, the mount would probably do petty things to get back at the PC. Real horses are petty like this. They'll step on your foot, or bite you when you're not looking, or pin you against the stall if you annoy them. They're brats.
If a PC were to polymorph into a horse, they can do any damn fool thing they want. If they want to run up and bite some guy's arm and drag him down and sit on him, they can.
I guess my answer is: Can a horse start a grapple? Yes, it is physically capable of doing so. Would a horse start a grapple? No, it is mentally disinclined from taking that action.
If you're absolutely dead set against allowing this in any form, you can simply rule that only primary natural attacks can be used to initiate a grapple unless the monster entry specifies otherwise. Since hoof is the primary attack and hooves can't grip, there's no possibility of starting a grapple.