With proper aim can an Ambassador always one-shot a sniper?

The ambassador does 102 damage with a crit (headshot only). If you shoot and don't wait for the target receptacle to shrink back down your next shot won't crit, even if it hits in the head. So in short, you can't kill a sniper in one hit with a headshot unless he has 102 health or less.

So if you want your achievement you have to pick on snipers with low health, shoot from a far distance where they won't see/look for you (like behind them) or soften them up with a body shot, wait for the target receptacle to shrink back down to normal and headshot them that way. The third option is the hardest because a good sniper when shot once will generally move and look around.

My suggestion, go cloak and dagger and get behind their lines. Find a sniper off on his own and shoot him in the head from a far distance. There is no damage drop off over distance for crits (like a headshot with the ambassador). A second option is to find the hurt snipers and shoot them that way.

For a good read: http://tf2wiki.net/wiki/Ambassador

Long range ambassador headshot:

Dycker covered most of it, but also be aware that if someone has the full Croc-o-Style set on that they cannot be killed with headshots. The shield and rifle are pretty common, so just look for snipers with a Croc on their head.

Two things to add:

1: If you're really just going for the achievement, you COULD ask a friend to help you out, either in a private server, or on another team. This is kind of cheese though, so if you're not comfortable with that, use Dycker's suggestion.

2: If what you're really concerned about is taking out a Sniper, just fire more than once. After a single headshot, another one or two hits will take the sniper down.

Neither of these are the real 'best' answer, but both are worth considering if using the Ambassador.