They can. Horses can be allergic to pollen, just the same as we can - I even know a horse that I suspect may be allergic to the same kinds I am.
Respiratory allergy symptoms are rarer in equines than breaking out in hives.
There are no good antihistamines for horses - treatment consists of corticosteroids or desensitization shots (if the allergen can be identified).
Increasing, reducing, or changing the time of turnout can also help.