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 simple answer is that they would prevent capitalism from occurring by not enabling it to begin with.

It should be understood that the only practical difference between anarchism and anarcho-capitalism is which claims to ownership the society is willing to enforce and which it consider superfluous and thus ignores. The standard anarchists propose for deciding this is that land and infrastructure not being used or occupied by the claimant cannot be considered the claimant's rightful property. This means a merchant or artisan could run their own small business, but that they wouldn't be able to eventually become the owner of a thousand-store franchise that other people run on their behalf. As if often the case in radical politics: scale matters.

What difficulties might this would-be capitalist encounter while trying to establish herself in a theoretical left-anarchist society? No, there wouldn't be a secret anti-capitalist police force to come after her in the night, but there probably wouldn't be many people (if anyone at all) willing to agree to the terms of her operations, and her society probably wouldn't be willing to enforce her claims to ownership of things outside her direct sphere of influence.

But if I buy something, that means that thing is mine. How can my society just choose to ignore that claim?

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. 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 have some power, but they typically can't override the laws of the land. In my country, for example, it doesn't matter how many contracts I sign saying my body is now the legal property of someone else, because if they took those contracts to court, the courts would refuse to enforce it, because a contract between two people doesn't change the fact that slavery is illegal.