Depends what you mean by conditioning?
For me, the idea of "conditioning" my dogs is that I'm getting them to a level of fitness which they need to be able to compete. That's physically able, and mentally able. If I was watching an agility ring and saw an overweight dog who was clearly stressed and unprepared for a show atmopshere, I would think that it needed some more conditioning to be prepared for the show, regardless of whether it can complete all the obstacles etc. (Of course, you also see very well conditioned dogs who are physically and mentally fit, but aren't trained very well at all!
With a puppy, I would gently introduce them to the show atmosphere and crowds of dogs and people from an early age - mental conditioning. I would also gradually build up the length and distances of walks, and probably introduce hydrotherapy and chase-games once they were around 9 months old (depends on the pup though) for physical conditioning. I guess that would start at 8weeks old (?) but it would be age appropriate. I'd prefer to start conditioning gradually from a younger age, rather than wait until the dog is 12months old and then start doing agility when it's not fit. Long before I'd start agility training I'd do some agility-specific exercises to build up the muscles that are primarily used in agility, and to increase body awareness and balance, so that when my dog meets the dogwalk or long jumps, they can meet the obstacle without falling off or pulling a muscle.
Anything else is just training (for me!). And I'd start training a puppy for agility or flyball as soon as I got it, regardless of age. At 8 weeks old, that would be training a sit/down/wait/recall, but that's still training for future agility.