I think it only happened once in my coaching career (12+ years as head, assistant coach in youth hockey) where a player who played for us the prior year was injured for tryouts. He broke his arm playing lacrosse just before tryouts. Luckily, he was the 2nd (maybe it was 3rd) leading scorer on the team the year before. We got approval from the board of our organization to offer him a spot without skating at tryouts. The organization still made his family pay the tryout fee (to get him in the registration system, they said) and he showed up and was on the bench during the skates. No one complained (that I know of) that he got a spot because he was known to be one of the better players.
I doubt we would have taken an injured player from another team without trying out unless that player was known to be exceptional. If the injured player had been in the bottom half of the team, that would have made things much more difficult - probably would have been a judgement call based on who showed up at tryouts. So, yes, call the coach and let him know and go from there.