These responses aren't really grasping the problem. I can't keep the API key on the server side if the submit action needs to send the API key to the remote API. The only way I could do that is if I created a server-side wrapper that handled the API call. How do I do that with Gatsby?

Given that scenario I think you have (at least) two options:

• Use serverless functions to proxy the call without having to host a server (where those environment variables would be stored). Depending on your hosting platform you can use Netlify functions (for Netlify hosting), Gatsby functions (for Gatsby Cloud hosting), AWS Lambda (for Amazon-related hosting), Google Cloud Functions, etc. You can even use them without hosting the project on these platforms.
• Proxy the API calls through a backend-for-frontend, like an Express server, and add the secrets to your Express server

Both approaches rely on the fact that the serverless function/Express server will be used as middleware of those requests, hiding your secrets from the browser.