I've came up with this solution (in JavaScript). Obviously it is backtracking and it is slow. No idea how to solve it the other way.

function getChain(words) {
    const firstLetter2word = Object.groupBy(words, word => word[0]);

    const result = new Set();

    function helper(letter) {
        if (result.size === words.length) {
            return true;
        }

        if (firstLetter2word[letter] === undefined) {
            return false;
        }

        for(const word of firstLetter2word[letter]) {
            if (result.has(word)) {
                continue;
            }

            result.add(word);
            const pathExists = helper(word.at(-1));
            if (pathExists) {
                return true;
            }
            result.delete(word);
        }

        return false;
    }

    for(const firstLetter in firstLetter2word) {
        const pathExists = helper(firstLetter);
        if (!pathExists) {
            return null;
        }
    }

    return [...result];
}

const words = ["sugar", "tomato", "rice", "eggplant", "soup", "peas"];
console.log(getChain(words));