You can and are supposed to report self-employment income on Schedule C (or C-EZ if eligible, which a programmer likely is) even when the payer isn't required to give you 1099-MISC (or 1099-K for a payment network now). From there, after deducting permitted expenses, it flows to 1040 (for income tax) and Schedule SE (for self-employment tax). See https://www.irs.gov/individuals/self-employed for some basics and lots of useful links.
If this income is large enough your tax on it will be more than $1000, you may need to make quarterly estimated payments (OR if you also have a 'day job' have that employer increase your withholding) to avoid an underpayment penalty. But if this is the first year you have significant self-employment income (or other taxable but unwithheld income like realized capital gains) and your economic/tax situation is otherwise unchanged -- i.e. you have the same (or more) payroll income with the same (or more) withholding -- then there is a 'safe harbor': if your withholding plus estimated payments this year is too low to pay this year's tax but it is enough to pay last year's tax you escape the penalty. (You still need to pay the tax due, of course, so keep the funds available for that.) At the end of the first year when you prepare your return you will see how the numbers work out and can more easily do a good estimate for the following year(s).
A single-member LLC or 'S' corp is usually disregarded for tax purposes, although you can elect otherwise, while a (traditional) 'C' corp is more complicated and AIUI out-of-scope for this Stack; see https://www.irs.gov/businesses/small-businesses-self-employed/business-structures for more.
