It really depends on what we consider "reason" to be. If we use the verb "reason" we get this:
rea-son (rezn) v. intr. 1. To use the faculty of reason; think logically. 2. To talk or argue logically and persuasively.
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Excerpted from American Heritage Talking Dictionary
Copyright © 1997 The Learning Company, Inc. All Rights Reserved.
Now, for #1, well, we'd have to define what it means to think logically or use the faculty of reason. Depending upon how loosely or rigorously we define it, we could leave it open to rather stupid creatures, or limit the ability to only the most skeptical of minds(merely intelligent ones being prone to illogical self-delusion). After all, the very nature of an information processing system demands that it "thinks" logically, if only because few other systems than a semi-logical system could get life-sustaining results in the long run.
As for #2, well, no. Not all people have the ability, including the smart ones, as persuasiveness is the social skill of finding a common ground of common premises and working with those premises, and not all people can figure out the functionings of other people well enough to persuade them. However, assuming away the social problem, we still run into the issue of logic. How do we define "logically", as a lot of minds can carry with them problems that undermine their ability to be fully logical. This means that even intelligent people with high processing abilities could be disqualified because they carry hidden assumptions and other rather unproblematic logical flaws.... unless we define the term "logically" broadly in which case partially fallacious thoughts can be included simply due to their attempts at reasonably processing.
I suppose though, we could argue theoretical potential to do so, which would make things easier.
If we argued theoretical potential and based upon definition 2 so that it includes social and logical needs, then I guess we can argue that the mentally handicapped cannot reason. I am being way too analytical... perhaps because I am tired. I really do not know though.