Does having a permanent no running profile for an ankle injury automatically disqualify me from becoming an aviation warrant officer?

Rated career fields are always in demand, so you're competing against others who can run. Not sure about Army aviation, but AF officers lose their pilot qualification if they can't egress (run) from the aircraft during a ground emergency. You don't have to run more than 100 yards, and fire can be a great motivator.

Nothing is completely automatic the best way to find out is apply worst they can do is say No.

Unless it has changed WOBC will not except anyone with a profile. They did back when I became a warrant but stopped shortly after.