But in left-anarchism, what prevents people from engaging in capitalism? If I pay somebody else to do something for me, am I breaking some sort of anarchist legal code that would demand I be punished, even if that other person agrees to my terms?

The practical difference between left-anarchism and anarcho-capitalism is which claims to ownership the society is willing to enforce and which it considers superfluous and thus ignores. What anarchists propose is that claims to land or infrastructure not being used or occupied by the claimant be considered illegitimate. This means a merchant, artisan, or otherwise independent laborer could could their own small business if they prefer, but that they couldn't grow from that point into being the owner of a thousand-store franchise that other people run on their behalf (as if this happened, their claim to ownership of those stores would be rendered null).

So the only difficulties a would-be capitalist can be expected to encounter if trying to establish herself in a theoretical left-anarchist society is that nobody would be willing to consent to her terms and that the society wouldn't be willing to enforce some of her claims. This is far from what people who ask this sort of question often have in mind, which is that there would be some sort of anarchist police force that goes around arresting anybody who tries to purchase labor power on the black market.

But if I buy something, that makes it mine. How can my society just choose to ignore that?

Well, why would an entire society be forced to respect an agreement between two people? Ownership isn't a physical object that can be handed from one person to another, but a series of agreements between claimant and society. And just because a society agrees to enforce a claim to ownership made by one person doesn't mean it must also enforce any further claims to that thing made by anyone of that first person's choosing.

Yes, contracts do have some power, but they typically aren't capable of overriding the laws of the land. Where I live, it doesn't matter how many contracts I sign agreeing my body is now someone else's property, because if they took that contract to a court, the court wouldn't enforce it, because the existence of our contract doesn't change the fact that slavery is illegal.