This did not happen and there is no contemporary evidence whatsoever for it. The archers would not have been shooting with intent to incapacitate their enemies a day or so later when infection set in. They were shooting to disrupt the immediate threat of thousands of French soldiers advancing on their positions. When they wanted to ensure that a wounded enemy would not be able to get up to fight again, the English finished them off with swords, axes, and knives rather than wait around for a possible infection to do the job.

Sir John Smythe, writing almost two centuries after Agincourt, did mention in his Certain discourses that he had heard varios French captains and gentlemen blame English victories in the past on the use of poisoned arrows. However he considered this "a wrong opinion". Instead he argued that it was simply bits of rust from the arrowhead which got in the wound and prevented it form healing, rather than any intentional poison.

I'm not familiar with contemporary accounts, but I would suggest that, even if they were aware of the effects, diarrhea would not be seen as necessary since a rusty arrowhead is already pretty good at causing an infection on its own. Also, as MI13 points out, causing infections days after the battle generally isn't an immediate concern.

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