Vtyjkyuk

It has a heavy mass inside of its body which it can twitch/move vigorously.

It would be somewhat analogous to a mexican jumping bean:
https://en.wikipedia.org/wiki/Mexican_jumping_bean

These are seed pods that have been inhabited by the larva of a small
moth (Cydia deshaisiana) and are native to Mexico. The "bean" is
usually tan to brown in color. It "jumps" when heated because the
larva spasms in an attempt to roll the seed to a cooler environment to
avoid dehydration and consequent death.

The mimic could contain a mass which it could similarly "spasm", thus throwing itself around with its own inertia. (It would move in any direction in a series of hops, provided it has traction on the floor. By twisting the mass concurrently with a hop, it would be able to adjust the direction it is facing. On the other hand, I doubt it would have much luck with stairs.)

Edit / final note: Based on the depictions of mimics I've commonly seen, perhaps its tongue could perform this function?

The D&D version is described in great detail in Ed Greenwood's classic Dragon article "The Ecology of the Mimic", and, as with the article as a whole, the mimic's locomotion is a brilliant balance of ridiculously silly and seriously plausible.

I don't have access to the magazine,¹ but from memory:

Mimics are amorphous beings. They don't spend all day looking like a chest; they see or hear adventurers coming, and mimic a chest because they're smart enough to know that's how you catch adventurers.² But they're not "some sort of slime-thing that's using a chest as a shell", they actually form their hides into rectangular shapes that look and feel as tough as wood, metal, or stone.³

They can rapidly form and extend pseudopods. Hence the famous Monster Manual drawing:

They can also secrete and dissolve a strong adhesive at will.

The way they move is by throwing out a pseudopod, covering it with glue, dragging their body over, and dissolving the glue as they toss out another pseudopod for the next "step". They can move about as fast as humans this way. This also means they can travel on walls and ceilings as easily as on floors.³

Also, most people forget that D&D mimics are intelligent, and will usually try to con the players through fast-talking instead of attacking them. And, while this is probably even less relevant here, they're immune to alcohol but have been known to feign drunkenness to trick adventurers.

So, how's that for an image: a mimic drunkenly dragging itself around the floor and the walls pseudo-fist by pseudo-fist, trying to make a deal with the PCs.

_{1. It really is worth reading, but I don't know where to find it. I mean, I'm sure you can find a pirated copy somewhere online pretty easily, but I don't want to encourage that. Go to the library, maybe?}

_{2. They mimic other things in other environments. A statue in a city so they can eat people tossing coins in the wishing well, a bundle of war pennants in an orc camp, etc.}

_{3. And how do they decorate themselves to have the right color and texture? I don't remember the details, but there were details. Something to do with pigment sacs and a special circulatory system, and also multiple rapidly growing layers of skin.}

_{4. Yeah, that last part doesn't make sense—it should be harder to pull yourself up than sideways—but then AD&D never really got gravity.}

How about it's a hermit crab-type deal? Crabs legs and mouth pieces are pretty uncomfortable to look at, probably because there are so many bits moving around. If it's ambushing people who try to open it up then it makes sense for it to have a good reach and fast hands, like a boxer crab. I really don't like crabs.

If it has big claws that reach out to grab whoever opens it up, it would have to fling the lid of the chest back really fast to counteract the momentum of it reaching forward, right?

To move itself around its legs would have to be close to the opening on the chest too and this thing would tip the chest on its longest axis to scuttle from side to side. Tipping would be difficult if it has stumpy limbs, so these appendages would have to be long and/or chunky.

Being geared up to scuttling side to side is practical if you're a chest because that'd be pushed up against a wall unless it's a feature chest or the interior designers don't use the space in the room efficiently.

That wouldn't make this thing great at chasing but it's an ambush predator anyway. Unless you're in a dead end of a corridor.

As far as the chest goes, that'd just be whatever's around, evolving a shell that looks exactly like a chest is a long shot, especially if chest-fashion in this world shifts regularly like it does in this one. It'd take a lot of very specific genetic changes to form a shell that looks just like a chest and there would be a lot of really messed up looking chests that weren't 100% right if this were how the chests are being formed.

These things might be laying a whole bunch of eggs so the smaller ones might be catching small animals from like empty cans or cat flaps or whatever. Any hole they can get into I suppose. You seen the movie teeth? All viable options.

I guess these things would have to lie in wait for really long periods in the abandoned buildings to avoid being fumigated or something. They'd have to be really energy efficient to survive, or fed by whoever's using these things as a security measure.

Maybe to stay tight in their chest, vase or whatever they're wearing, this things should be secreting an adhesive from its abdomen/tail. It's probably worth noting that out of a shell, hermit crabs look a bit like a creepy hand.

What's the story about?

The mimic can transform itself into something that someone else will move to the desired location

A crate mimic might always be stuck as some variety of crate, but it can still switch up its label. Mimic wants to go to the big city? Add a mailing address. Mimic wants to lay low for awhile? "Industrial Waste - Do NOT Open" and boom, you're in the local swamp.

It has legs, but not connected to any externally visible hips.

It can fold its legs beneath itself, so that in a sitting position the legs are completely hidden by its four sides. Just like a sitting chicken.

It also had many, many legs that move in a centipede-like motion. Meaning it can run surprisingly fast for its shape and size.

It will also follow its master wherever they go, even in time and into the outer dimensions. And you absolutely don't want to see these things mating.

My suggestion is similar to abarnert's, and I suspect it might be influenced by having read Ed Greenwood's article at some point in the past. But it's got it's own little twist to it: instead of using a sticky pseudopod to move around, what if the sticky part was its tongue? So when closed, looks like a normal treasure chest. When it wants to move, it opens the lid, and the hidden, sticky tongue pops out. Like a frog or reptile, the tongue is very stretchy, and the inner mimic is mostly just tongue. It would shoot its tongue at a location a few feet away, anchor itself with, perhaps, special barbs designed for this purpose, and then drag itself forward. Detach tongue, repeat. The other use for this stretchy, sticky, barbed tongue would, of course, be ensnaring prey that wandered too close.

Ooh, or instead of pulling with the tongue, it could push! Then it would look rather like a clam "walking" around.

I like the idea of a chest-like monster which hides its arms/legs inside itself.

When he feels like moving, he just needs to put them outside its "mouth"; the idea would be for it to look and move like a spider in this "walk/run instance".

This also gives it the ability to actually grab its preys once they are close enough.

I know you've accepted an answer, but I might as well give some of my thoughts too.

It's sessile: be it an animal or a plant, it needn't move at all. It could root itself to the ground and simply wait for a passer-by. When the human - or some other curious animal - touched its exterior (Which could be covered in mechanosensory trigger-hairs), its jaws would open and snap shut, severing the person's arm and digesting his or her hand.

It's a gastropod: or, at least, it moves like a gastropod. On its underside, there is a muscular ventral foot - not like ours, but like a slug or snail's. The foot would be lubricated with mucus and covered in cilia (tiny hair-like organs), and the Mimic would move by way of waves of muscular contractions of the foot.

It's a slime mould: again, I'm not suggesting that you place Mimics taxonomically as pseudo-multicellular microbes, but at least that they move like a colony of one. Basically, slime moulds move by swinging their gelatinous mass back and forth slightly, and then extending tendrils, pulling them across the ground. This movement would be very slow and hard to observe.

It's an echinoderm: Echinoderms, such as starfish and urchins, have ventra which are covered in tiny, hydraulic tendrils called "tube feet". They are, essentially, used like we would use our feet, except that they are so small that they can barely be seen without looking closely.

It's a ricochetal saltatore: well that's a mouthful. Basically, it would have a single leg (Stored inside the "chest" when not necessary), and hop like an Eponan Springcroc. Kangaroos released stored energy with each jump, which helps them jump further.