Actually the question "Why did the chicken cross the road?" is perfectly valid. While you may be correct that the chicken does not have a 'reason' for crossing the road because reasons (used precisely rather than as in common parlance) require intentionality with regard to their object, causes do not require intentionality and yet are at least as commonly if not more commonly the object of the interogative 'why' as reasons are.
To put it simply, I may say that the cause of the chicken's crossing of the road was the action of a particularly strong gust of wind in that direction. This provides an adequate explanation for the phenomenon and answers the question "Why did the chicken cross the road?" without imputing sentience or intentionality to the chicken's actions.
I may further say (if I wish) that the chicken crossed the road to eat the grain on the other side. This both imputes intentionality to the chicken, adequately explains the phenomenon and answers the question "Why did the chicken cross the road?" But wait, you may be saying, you just told us that intentionality isn't necessary to answer the question. I did say that and I stand by it, but that does not mean that intentionality may not be involved in the answer to the question. In this case, however, the intentionality while needed to answer the question, is only tangentially related to the effect under examination. Specifically to answer the question "Why did the chicken cross the road?" we are pointing out that the chicken intended to consume a certain pile of grain, and that the road was between the chicken and that pile. We still have not imputed to the chicken any knowledge of the road "as a road". Rather we have simply explained the conditions and the intentions which led to the action of the chicken crossing the road, whether or not the chicken had a full understanding of those conditions.
Finally we must address the standard answer to the question: "To get to the other side." Again this answer imputes intentionality to the chicken's actions (the chicken did it 'to get' something) and it seems to imply a knowledge of the road (to understand 'the other side' the chicken must have knowledge of some object with two sides, understand that it is on one side of said object and desire to cross the object to reach the other side). Implied in this answer is that there is no further motivation other than getting to "the other side" and hence we cannot suggest that the answer simply left off the fact that there was a pile of grain on the other side which is the 'real' reason the chicken crossed the road. No. The chicken must have crossed the road for the sole and ultimate purpose of reaching the other side of "the road". How are we to reconcile this with the (most unassailable) assumption that the chicken has no knowledge of the road "as a road" and the need to allow this statement as a positive answer to the question "Why did the chicken cross the road?" We have specified that the chicken has no knowledge of a road "as a road". However, we have never suggested that the chicken has no knowledge of the road "as something". What then is the nature of the road as the chicken perceives it? We would not be unjustified in suggesting that at the very least the chicken has access to its own sensory data. It then must have a knowledge of the road as the "extended-hard-flatspace". We need go no further in our suppositions. We have here a chicken with an exploratory bent who wishes to discover what lies beyond the "extended-hard-flatspace". This adequately explains the phenomenon, assigns to the chicken a state of intentionality, relates that state of intentionality to the road, and answers the question "Why did the chicken cross the road?" with the statement "To get to the other side" all without in the least requiring that the chicken understand roads in the sense that we as humans understand roads.