WordPress itself can communicate with external API’s through wp_remote_get and wp_remote_post. See the codex.
In your case, you could create a page template called page-search.php (keep in mind that search.php is reserved for the internal WordPress search functionality, so you would define it as a template, and select it when creating the page), and run your wp_remote_get code in that template.
There are of course other ways of doing it, like utilizing Javascript, with or without any of its available libraries.
The fetch API makes it quite easy to do it with Javascript.
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