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Just use an alien.

If it has a large brain you can't get around sleep, every organism we know of with a brain that is enlarged (complex at all) sleeps. there is just no escaping it. As long as you use earth life you have no options. But we don't know why we need it, which leaves you an out.

If your lifeform evolved on a different planet they may not need sleep, you can literally just say it has drastically different neurochemistry/biochemistry and does not need sleep and no one can argue since we don't know why earth life needs it.

Polycephaly is a medical condition, in which an animal has multiple heads. This is a special type of conjoined twin. 2 is the most common form, but there is at least 1 case of 3 heads documented (a turtle).

In many cases, each head is capable of sleeping independently of the other, allowing your Biological Creature to function without turpor.

In some cases, each head controls a different part of the body (e.g. Abby and Bittany Hensel, born 1990, each control one hand and one leg. Interestingly, if one of them develops stomach ache, the other twin experiences the sensation instead).

In other cases, both heads are able to control all-or-most of the body - however, when they disagree on an action, this can cause the body to "lock up" and paralyse.

What you require is a situation where both heads can control the whole body, and sleep independently.

You don't need new organs you just need a few tweaks to the way the brain operates & Dolphins have already solved this one for you, obviously they sink if they stop swimming & if they sink they drown so they can't sleep, well they do, but they sleep one hemisphere of their brain at a time.

There's some evidence that dolphins can display slightly different personalities depending on which half (or none) of their brain is sleeping, which is interesting but perhaps not relevant :)

Ah! but on a re-read I see you don't want that option.

Well there's also a commercially available drug in the US that stops you needing any sleep, there was a documentary I saw on it a few years ago.

I can't remember what either that drug or the documentary was called but you might find this article helpful.

Sleep is largely a function of biochemical process in the brain so simply manipulating the genes to adjust the Adenosine receptors to prevent uptake of this neurotransmitter or reduce or else prevent production of it might do the trick.

A downside is that sleep is an integral part of transferring short term memory to long term memory & as far as we know no organism that uses long term memory can do without it indefinitely.

I don't know 'why' the brain needs sleep to transfer short term memory to long term but as sleep is a such a vulnerable state to be in there must be strong evolutionary pressure against any alternative to the dolphins for it not to already have appeared.

Long & the short is the dolphin answer seems the best option to get the 'appearance' of what you ask for despite the fact you remove that option in your question.

But there's no reason to stick to the two hemisphere plan of the the Dolphins (presumably shared by whales).

You could split the brain into four or more with overlapping sleep patterns such that each part (eventually) shares sleep time with each other part, you can use that to plausibly say long term memory is duplicated to all parts of the brain during this time to avoid any long term variation in memory & personality during different parts of their sleep pattern.

Brain with three hemispheres

The state in dolphins and some birds that could keep some functions while sleeping is achieved by working with only one of the hemispheres of their brains while in "sleep state" the other part is resting.

If you add an extra hemisphere that would support the "awake" part with their normal functions you could cycle between them to achieve a "non-sleep state" during most part of the life of the creature that you are designing.

While multi-brain and multi-hemisphere answers have been posed, they all result in some loose of function while one part of the cognitive system is down. This seems against what the OP is looking for since he does not want dolphin style torpor.

However, computer scientists have already devised a number of systems called RAID controllers to be used by servers to replicate data across multiple hard drives in case of a disk failure and to allow for scheduled maintenance without downtime. When you consider sleep as analogous to server maintenance, it becomes obvious that a brain could use a similar system to overcome the need for sleep with a somewhat larger brain (or rather 2 identical brains). By replicating functionality, when one brain is ready to sleep, it wakes up the other brain duplicate it's memories of the day to it, and then goes to sleep. In this fashion, you would not notice a performance drop-off or change in personality because both brains are doing the same things with the same hardware and the same information, just at different times of day.

As jdunlop points out in the comments below, there are two possible concerns here: One being that sleep is poorly understood, and the other being that consciousness is poorly understood.

As far as those concerns go, the nature of sleep is irrelevant. We do not need to understand much about how sleep works to understand that it is an "offline process" where the brain is doing something other than what it does while you are awake. Whether it is a time to sort memories, replenish neurotransmitters, repair damage, etc. is inconsequential. As long as it is an isolated enough system to do what the brain is designed to do while sleeping, it can do that while the second brain takes over.

As for how consciousness goes, this is a more existential question than anything. To the outside observer, two brains attached to the same body with all the same capabilities, memories, skills, endocrine system, etc. would be indistinguishably the same person. If you ask either brain the same question, they will generally come to the same answer as long as they are syncing up their memories every switch over, and neither of them is significantly damaged.

From the inside, you would technically only "be you" half of the time, but you would always receive all of the memories and experiences as the other you. When "you" are in control, you really would not be able to tell the difference of when were awake, and when you remember being awake. You might experience something like deja vu twice a day for those moments when both minds are trading off, but the human mind is full of filters designed to merge multiple inputs into a single stream of consciousness; so, a species designed to function this way would likely have a mechanism for making the switch over feel seamless the same way we only see one image when we look at something with two eyes.

[Fiction]

Sleep is a natural stress reliever. In humans, and even in animals, you can expect a lack of sleep to lead to a situation of Hypertension. So, any sentient animal species that doesn't sleep at all has to be really stressed, which means its lifespan will be short lived.

Next, not sleeping is akin to an extreme case of insomnia which is biologically evolved in your case.

Since insomnia is a common occurence in humans, you can look into conditions that can trigger it/ break the regular sleep cycle, and factors that can keep it broken.

For example: In below, the bodily chemicals/ hormones work in tandem to affect the regular sleep in humans:

Exposure to light stimulates a nerve pathway from the retina in the eye to an area in the brain called the hypothalamus. There, a special center called the suprachiasmatic nucleus (SCN) initiates signals to other parts of the brain that control hormones, body temperature and other functions that play a role in making us feel sleepy or wide awake.

Melatonin is a natural hormone made by your body's pineal (pih-knee-uhl) gland. This is a pea-sized gland located just above the middle of the brain. During the day the pineal is inactive. When the sun goes down and darkness occurs, the pineal is "turned on" by the SCN and begins to actively produce melatonin, which is released into the blood. Usually, this occurs around 9 pm. As a result, melatonin levels in the blood rise sharply and you begin to feel less alert.

So, all you need to do is establish interference in a similar system/ remove the hormone, and your fictional species can go sleepless.

Next, give this species a short life, and an early reproductive age, where the stress they have on themselves doesn't affect their reproductive abilities by much yet.

Sleep is mostly required for highly intelligent lifeforms or lifeforms with complex memory functions as far as I know (Bacterias don't sleep). This is different from physical rest, which is for conserving energy. As such a life form with two independent brains that take turns, would probably not need to sleep, since each brain sleeps half the time in turn, despite the creature itself being awake. Both brains need to have identical functions, the only problem being sharing/syncing of memories between the two. Otherwise you will end up with a race with dissociative personality disorder. As for chemicals to prevent sleeping, that's just overdrafting, the brain will eventually shutdown , maybe even permanently.