If we ever have ewes in to lamb we use the type of half-round horse bucket which hangs on the bars of hurdles, and position it above the height of backsides but just low enough to drink out of.
For outdoor lambing, we take a bucket out to each ewe where she has lambed, as she will tend to stay there for a couple of days, but we place the bucket a bit away from the group. We also have ball-cock drinkers in each paddock, but so far no-one has lambed into one of those
Yes, they can lamb into a bucket on the ground and even if the lamb doesn't drown it can die of hypothermia