Based on Google’s documentation for Sitelinks Searchbox you need to start by creating a website data item for your homepage. From there you can use the potential action field to create a search action data item. You will want your website data item JSON-LD to look like this:
<script type="application/ld+json">
{
"@context": "http://schema.org",
"@type": "WebSite",
"potentialAction": {
"@type": "SearchAction",
"target": "https://www.example.com/search?q={search_term_string}",
"name": "Search Example Site",
"query-input": "required name=search_term_string",
"@id": "https://www.example.com/#SearchAction"
},
"url": "https://www.example.com/",
"name": "Example Website",
"@id": "https://www.example.com/#WebSite"
}
</script>
This is what your search action data item will look like in the editor:
To verify the work that you’ve done, use the view json-ld feature within the editor. Then copy the markup and input it as code into Google’s Rich Results Tool.