It might be possible...
No arthropod has ever evolved from an exoskeleton to endoskeleton body plan, but that is not the same as saying it could not happen given the right circumstances. There is one group of exoskeleton based organism that have made this transition which at least suggests it might be possible: Coleoidea.
Coleoidea are a sub-class of cephalopods where the exoskeleton has been moved inside of the the body. This group includes octopuses, squid, and cuttlefish. When you look at their ancestry they basically evolved from worms to snails to nautaloids to coleoidea. So, while the hard shell seems like it would have been there to stay, being that a protective shell should be better than no shell, its niche called for flexibility above all other features.
Likewise, an arthropod which evolves under enough pressure to become more flexible may also loose its shell. At first I would suppose that the arthropod would develop a progressively softer and softer shell for squeezing though hard to reach places until its shell becomes so soft, it is almost skin like, and your arthropod would move more like an octopus than a bug. However, it may want to retain bits of harder shell in selective places: mostly the mouth for chewing food and close to vital organs to protect it from accidental death.
Over time the size, shape, and position of these selective plates will be optimized. And for an organism that values squishiness, the most optimal configuration for these plates is as small and close to the vital bits as possible. So, like the Coleoidea, your "softopods" would pull their bony bits inside of the body over time. Once its outsides have become its insides, future generations would become able to start using those plates as attachment points for musculature that could eventually become a proper endo-skeleton.
But this is probably not useful to you
The evolutionary path to get from an arthropod to an endoskeletal creature would be very long; likely hundreds of millions of years. In this amount of time, it would not just be the skeleton changing, but everything about the creature would change. By the time you get from point A to point B, the new species will be so far removed from arthropods that is will not even be recognizable as such. You will probably have a creature that has convergently evolved to be far more coleoidea like than arthropod; so, the simpler conclusion is to not go down the the whole arthropod evolutionary track at all, but instead plan to start from somewhere else.
My guess is that your goal is really just a terrapod like animal with the extra appendages. Believe it or not, this is easiest done by just starting with a terrapod and adding the extra appendages. While the extra appendage thing has never been successful enough to catch on, the mutation does happen often enough in nature that it could catch on given the right evolutionary pressure.
For large terrestrial creatures, limbs are a big investment in weight and resources; so, going from 4 to 6 or 8 is a huge sacrifice that does not benefit most niches, but there is one niche that has only recently evolved where this sort of mutation could be helpful: Tool Users. While a frog, cat, dog, lizard, or bear would only be slowed down by extra appendages, there are 4 basic body plans in works of fiction where a tool using species would do very well to have 6 appendages instead of 4 making it possible for a random mutation resulting in extra limbs to catch on:
Gegeinoid: This body plan is where a bipedal tool user has extra hands to work with. While humans have gotten along just fine with 2 arms, we often find ourselves limited by our lack of hands. Many things we try to do would simply be much easier with extra arms; so, if such a mutation were to happen in a stone-aged civilization, the advantage could be enough for that person to advance to the top of his clan giving him preferential matting rights allowing the mutation to spread to a whole population over time.
Centauroid: This body plan is where a bipedal tool user gets an extra set of feet to work with. While these feet would not automatically be bovine as often depicted, quadrupeds can generally move much faster than bipeds; so, if this mutation where to happen in a place like the Asian Steppes where horseback riding became such a way of life, this mutation could take horses out of the equation and allow tool users to directly be able to move more quickly.
Dragonoid: Many birds are basic tool users, but they lack the dexterity to manipulate things with their talons. This is largely because their talons need to remain specialized enough to be feet that they can not also be fully specialized hands. If a bird were to sprout an extra set of legs, it could then split their functionality to have separate hands and feet improving their potential as tool users. Especially if their new hands are far enough forward to be able to better visualize what they are holding.
Veloxi: I'm not sure what else to call this one since the only place I can think of seeing this body plan is from the old 1980's game called Starflight. The veloxi were a mantis like race described as having one set of appendages that were exclusively hands, one set that were exclusively feet, and a middle set that could function as both, in theory giving it advantages similar to both the Gegeinoid and Centauroid body plan.
