With care, (if you can call this "care") a strong, hardy daylily can live in water for a long time. I've held some for two years in a bucket. On a lark, I've set and harvested a couple of pods from one of them.
A few tips for an extended stay in water;
They need good light. Shade will weaken the plant.
I use enough water to just cover the roots.
The water should be kept as cool as possible, so staging is important. (Bucket shaded, leaves getting strong light).
A little food occasionally is good. I use a weak hydroponic (complete) liquid fertilizer* and add a Cal-Mag supplement, because most fertilizers don't contain calcium.
Weekly water changes, maybe every three days, depending.
The roots will develop oddly, the larger tuberous roots will eventually rot, and, if you leave them for longer than a month you will lose a few, but the hardy, healthy plants will survive. When it's time to plant a daylily that's been in water long enough to affect the roots, it's in a fragile state, so give it a spell in a spot with mid-day shade until it starts to show new growth. That's usually a sign that new roots are growing.They'll not be as strong as if you had just planted them on arrival, but sometimes we do what we have to do. Everything is a compromise.
My favorite method for holding new arrivals is to heel them in or pot them up using fine landscape bark. Or you can sit the plants on the ground and cover the roots with the bark. Bark is far cheaper than potting mixes, and actually seems to grow daylilies better. You should see the roots on daylilies grown in bark.
It's also just as easy to add bark to that same bucket when you put the plants in it. Fill it with water afterward to thoroughly wet the bark, then tip to drain. Feed as above. Some of the healthiest looking plants I grow were potted up using nothing but bark.
*My fertilizer of choice is Dyna-Gro Foliage-Pro 9-3-6
