dmlittle's concern is a valid one and for most other types of API I would definitely agree it's not the right approach.
I still think it's reasonable for my use case but perhaps I should add another auth scheme as an optional alternative for the user who is concerned about their key potentially being caught in logs.
Your point about the documentation is also a good one - I should probably add a specific page just about the authentication approach. Added to the to-do list! Thanks.
FWIW you can add HTTP Basic Auth information in URL links and all major browsers and other HTTP clients (for the most part) should interpret it correctly.
https://:[API_KEY]@v6.exchangerate-api.com/v6/latest/USD
If you only have an API key and not a token (username) and secret (password) I recommend passing the API key as a password as some logging solutions do log the basic auth username in the data recorded.
But it seems like in this case it's mostly a rate limiting and identification exercise and not a secure protection of user data so the impact of exposure is substantially lower. So it does seem reasonable here.
Though I hope that OP has documented all over the place "do as I say not as I do" so people don't copy this pattern.