Here's an example of the API call that WikiData's search box uses for suggesting entities. The suggestion is based on the text "ch" in the search URL query parameter.

gistfile1.txt

curl --silent 'https://www.wikidata.org/w/api.php?action=wbsearchentities&search=ch&format=json&language=en&type=item&continue=0' | python -m json.tool
{
    "search": [
        {
            "aliases": [
                "encyclopaedia",
                "encyclop\u00e6dia"
            ],
            "description": "type of reference work",
            "id": "Q5292",
            "label": "encyclopedia",
            "url": "//www.wikidata.org/wiki/Q5292"
        },
        {
            "aliases": [
                "Encyclopaedia Brittanica",
                "Encyclopedia Britannica"
            ],
            "description": "general knowledge English-language encyclopaedia, first published in Scotland in 1768",
            "id": "Q455",
            "label": "Encyclop\u00e6dia Britannica",
            "url": "//www.wikidata.org/wiki/Q455"
        },
        {
            "description": "Wikipedia disambiguation page",
            "id": "Q1340122",
            "label": "Encyclopaedia",
            "url": "//www.wikidata.org/wiki/Q1340122"
        },
        {
            "aliases": [
                "Encyclop\u00e9die, ou dictionnaire raisonn\u00e9 des sciences, des arts et des m\u00e9tiers"
            ],
            "description": "general encyclopedia published in France between 1751 and 1772",
            "id": "Q447",
            "label": "Encyclop\u00e9die",
            "url": "//www.wikidata.org/wiki/Q447"
        },
        {
            "description": "a letter, usually treating some aspect of Catholic doctrine, sent by the Pope and addressed to the Catholic bishops",
            "id": "Q221409",
            "label": "encyclical",
            "url": "//www.wikidata.org/wiki/Q221409"
        },
        {
            "description": "free, online collaborative encyclopedia intended to document all of the 1.9 million living species known to science",
            "id": "Q82486",
            "label": "Encyclopedia of Life",
            "url": "//www.wikidata.org/wiki/Q82486"
        },
        {
            "id": "Q938726",
            "label": "Encyclopaedia Metallum",
            "url": "//www.wikidata.org/wiki/Q938726"
        }
    ],
    "search-continue": 7,
    "searchinfo": {
        "search": "encyc"
    },
    "success": 1
}

Hey mate,
what about the API request when we click search ?
The full text search in the knowledge base ?
I am struggling to do that in a programmatic way!

Cheers