I have a player who was so badly injured in training camp that he probably won't be able to play the entire following season.

Now in week 5 of the season, am I getting an email from my coaches that there is an issue with this injured/disabled player about playing time!?

I find that a bit weird.
Is this WAD?

Sure, even an injured player can be upset that he can't play.
Morale will likely drop as a result.
If that's the case here, then the email should specific indicate a loss of morale due to..., not general comunicated 'issue playing' time.

Haha, that definitely sounds like one needy player! I'd toss that in a bug report to the dev team to see how they want that to be presenting in the game.