When do I claim revenue if I earn it in 2022, but don't receive it until 2023?

If you are a cash basis taxpayer, you report income in the year in which it is received (2023 in this instance). 

However, you might want to contact your client and ensure that the client is not planning to issue a 1099 to you for the 2022 tax year. If that were to happen, it would cause a mismatch between what you reported on your 2022 return and the tax reporting statement (and you would likely receive a notice to that effect from the IRS).

It depends on how you've set up your accounting.  I expect that you are on a typical calendar year cycle.

The income would be posted to 2023 if you are using Cash Basis accounting, and you use the calendar year as your fiscal year you would book transactions in the year they occur. 

If you use the Accrual Method, it will show as revenue in the year that you prepared the invoice (2022).