As far as I know, we do not know of any creature with a system to reattach severed limbs, because if you lose a limb, odds are that the same creature which separated your limb is going to finish you off. So if you want the creature to be plausibly "natural", you will first need to describe an environment in which it is likely to lose an entire limb, but also able to retrieve it mostly intact. Perhaps these creatures live in an environment with very sharp, glassy rocks which are easy to slip on, so that the main threat isn't predators, but rather accidental dismemberment.
Most creatures have an immune system, which is a population of mobile cells which travel to where they are needed in the body and attack stuff. At the same time, most creatures treat "repair" as: "close up blood vessels and grow new skin". However, there is no real impediment to developing a "reattachment system" which is similar and parallel to the immune system.
Of course, the main problem is alignment. I think expecting the limb to just attach and orient itself correctly is really stretching plausibility. So I would stick with a creature that has prehensile limbs and digits (i.e., primates). That is, you want the creature to be able to place the severed limb in approximately the correct location. Even so, "close" isn't really "close enough" to reattach small blood vessels and nerves.
For precision, you need some way to mark up the piping and wiring so the repair cells know what to do. Fortunately, biology is really just a giant wetware information system, and there are many, many molecules which will do the trick. Just about every cell in every creature expresses proteins on its exterior. The body uses these proteins to identify "self". Adaptive immune systems use them to identify "invader". The reattach system can use them to identify "blood vessel 4e7h". Yes, every nerve and blood vessel, every muscle, tendon, ligament, etc. can be coded with its own unique identifiers. However, you don't really need them to all be unique. You just need them to be "locally unique". The creature itself should perform most of the alignment visually. Then, a blood vessel just needs to be unique relative to the few other blood vessels within a few cm of it. Thus, tags can be reused around the circumference of a limb, as long as they are adequately separated. Also, different kinds of tissues can reuse the same tags, as the repair system will need to identify tissue types anyway.
When a tissue with one tag is placed next to a tissue with a different tag, it can cause a hormone release which triggers pain receptors. So when the creature is attempting to position the limb, it simply rotates and moves it around until it minimizes the "contact pain". Pain nerves will tell it when the alignment is off, but the closer it gets to proper alignment, the more the pain signal quiets.
We know that the adult body harbors pluripotent stem cells which can differentiate into any other cell type. So repairing virtually any tissue type in vitro is at least theoretically possible just using biology we know about today. Once the tissues are adequately aligned, these cells just need to go to work, taking on the proper tissue tags and rebuilding the severed connections.
New blood vessels and nerves form all the time, but perhaps the biggest challenge is muscle. While animals can grow new muscle from scratch, when a muscle is severed, I am not aware of any creature that can reattach its own muscle fibers to repair a break. I don't think it's biologically impossible, but we don't have any models for it. Fortunately, bones also reattach and heal on their own all the time, with nothing more than a splint or cast (and for wild creatures, not even that).
The last challenge is sepsis. Whatever got into the exposed ends of the severed limb will end up on the inside of the body once the limb is reattached. The creature needs to clean it as much as possible. This is going to sound gross, but the most obvious way to do this is to lick the wounds. For the detached limb, that is easy enough. For the detachment point, the creature might not be able to reach it with its tongue. Even so, it will need to lick its paw/hand/foot/etc. and try to clean the wound before reattaching. Once it does, the immune system will need to rally and attack the interlopers once they have the advantage.
While I can imagine such a system working for the loss of a single limb, I would imagine that blood loss and shock would prevent a happy outcome for multiple amputation. Of course, those limits are up to you, but real creatures which can "safely" lose a limb rarely lose more than one at a time.