A sentient magic item always controls its own abilities
The section of the DMG (p. 214) on sentient magic items (and the corresponding part of the basic rules) contains a passage relevant to your question:
Sentient magic items function as NPCs under the DM’s control. Any activated property of the item is under the item’s control, not its wielder’s. As long as the wielder maintains a good relationship with the item, the wielder can access those properties normally. If the relationship is strained, the item can suppress its activated properties or even turn them against the wielder.
So even if the item takes control of or possesses its wielder, it doesn't matter whether or not that possessed wielder can still access the item's abilities, because the wielder was never truly in control of the item's abilities to begin with. The item already had control of its abilities, and it maintains that control after possessing its wielder.
The Eye of Vecna is not a sentient item
Note carefully the wording of who possesses the user of the Eye of Vecna (DMG, p. 224; emphasis added):
Vecna tears your soul from your body, devours it, and then takes control of the body like a puppet
The Eye of Vecna is not itself a sentient item. It is merely the eye of the deity named Vecna, and this deity, not the eye, is what possesses the eye's hapless user. There is no reason that Vecna would not be able to use his own eye after possessing the newly soulless body of its user. One has to assume he has prior experience with it, after all.
Your homebrew artifact works however you say it does
Obviously, the reasoning I've given above for the Eye of Vecna is very specific to the nature of that artifact. But that is always going to be the case for artifacts. Artifacts are not just ordinary magic items that happen to be super rare and super powerful. They are a special category of magic item, distinct from all others, and furthermore each artifact is unique, as mentioned in the DMG's section on artifacts (p. 219; emphasis added):
An artifact is a unique magic item of tremendous power, with its own origin and history. An artifact might have been created by gods or mortals of awesome power. [...]
Characters don’t typically find artifacts in the normal course of adventuring. In fact, artifacts only appear when you want them to, for they are as much plot devices as magic items. Tracking down and recovering an artifact is often the main goal of an adventure. [...]
The point is, every artifact is different from every other artifact. Just because the Eye of Vecna works as I've described doesn't mean your artifact has to work that way. For example, you could have a circlet containing a gem that houses the soul of a powerful archmage, and when it possesses its user the circlet disappears and the gem permanently embeds itself it the user's forehead, destroying the item and effectively replacing the user with the archmage. Or it could be a sentient item that simply takes command of its user (similar to a Dominate Person spell) without killing them or destroying their soul, while remaining a distinct item that the user can continue using as normal (at the item's command, of course).
Remember, an artifact is a plot device first and a giant block of rules text second. So first think about what makes sense for the nature of the item as you've envisioned it, and then write the rules text to match it. (But if you think this is something likely to fall into the PCs' hands, you should consider having a way to undo the possession.)