Your question doesn't have a clean, simple answer, but it doesn't in an interesting way. I know that's not exactly what you want to hear; allow me to explain:

It's true that black people arrived in the Americas on slave ships. It's also true that slaves are, by definition, not allowed to have property rights. They may possess control over certain property that their enslavers legally own, but if someone stole it from them or they sold it without leave to do so then that would be an offense against the enslaver's rights rather than their own. So it makes sense to wonder when property rights kick in, as it were. This is a tangled subject because there is no one date and the hard legal distinctions don't always map to lived reality.

Property which enslaved people are given the use of is not, legally, theirs. But as a practical matter enslavers sometimes treat it as though it is. This can be as extreme as allowing enslaved people to live independently in exchange for regular payments, which is such an obvious contradiction of the logic of slavery that enslavers try to curb it. An enslaved person who can make their own money and live on their own isn't much like a slave. Rather they look worryingly free. Some of them were informally "freed" through such arrangements, especially as barriers to manumission increased. A less extreme version of this is what was nearly a fixed custom of Southern life: allowing slaves to conduct some business selling produce and other things they raised or made on their own time. That's by no means a right to do so, but either way it still looks a lot like enslaved people exercising the normal range of property rights.

The flip of all this is that it can end in an instant. Anything a slave "owns" is really owned by the enslaver if they choose to exercise their rights. There's no obligation on the part of an enslaver to recognize customary practice or way to force them to honor agreements made previously. An enslaved person could spend years painstakingly raising the money to self-purchase and then have the enslaver just pocket it. (There is a partial exception to this in Colonial Louisiana, where French law granted a right of self-purchase.) It can be downright ghoulish how this plays out.

William Craft was up for sale alongside his sister. Same auction. A man who lives some distance away buys him. Craft has money he's been allowed to save up from his carpentry work to buy his sister. It's a tidy sum, though not her full price. Begs the man to buy his sister too fearing they will never see each other again. He promises to work hard, pay the rest of the price as soon as he can, never run away or cause problems, everything he can think of. He offers the money. The enslaver takes the money and quotes the law to him: Since he owns William, that's his money William is trying to entice him with.

But that's enslaved people and you asked about free blacks. We don't know who the first free blacks in early British North America were, but among them is a man named Anthony Johnson. He was able to assert property rights (over a person, no less) in the 1650s. Early on in the Chesapeake and New England, the racial caste system has not hardened as much as it will and lucky people with the right friends sometimes manage to do quite well for themselves. Depending on what points about slavery one thinks most important, and how one parses the very limited evidence, Johnson might have been a freed slave or sold as a slave but kept as an indentured servant until his time expired.

So we have William Craft in the 1840s and Anthony Johnson in the 1650s. Both are black men. Johnson has property rights and Craft does not, at least in these stages of their lives. The distinction between them isn't a specific disability that British colonial or American law attaches to their skin, but rather one of their legal condition. There is an insistence in the law that people who appear black or otherwise qualify are presumed enslaved until proven free, but that presumption is much weaker in the North than the South. There may have been individual sorts of property that a free black person could not own by law or custom, like a place in the whites-only cemetery or one of the better pews in a church, but there was no global ban on blacks owning property even in the most enslaved states.

To the best of my knowledge, poverty enforced by and created through white supremacy ensured that many had little to call their own without any legal help. In South Carolina and Louisiana in particular, free black men occasionally owned their own plantations. (Their property rights, like Anthony Johnson's, included property in people.) In SC, those free black men tend to be blood relations to white enslavers who set them up and understand themselves more as of intermediate racial status "brown" rather than "black." Where free blacks are much more common, in the Upper South, such a status isn't usually permitted by the local whites.

All that said, the closest I can come to a straight answer for you is to point to dates of emancipation. The legal disability ends with slavery, but only for people hitherto enslaved. That can be as late as late 1865, when the Thirteenth Amendment is ratified, or as early as the first people being freed by possibly immediate emancipation in Massachusetts or New Hampshire -we're not entirely sure what happened there; there were court cases which appear on paper to have enacted emancipation but slavery was already in steep decline and it's not clear that people at the time understood them as doing so- in the 1780s or when the first enslaved people age out of gradual emancipation in the first decade of the nineteenth century.