How can I create the cartesian product of a vector of vectors?

First, I’ll show you a recursive version.

typedef std::vector<int> Vi;
typedef std::vector<Vi> Vvi;

// recursive algorithm to to produce cart. prod.
// At any given moment, "me" points to some Vi in the middle of the
// input data set. 
//   for int i in *me:
//      add i to current result
//      recurse on next "me"
void cart_product(
    Vvi& rvvi,  // final result
    Vi&  rvi,   // current result 
    Vvi::const_iterator me, // current input
    Vvi::const_iterator end) // final input
{
    if(me == end) {
        // terminal condition of the recursion. We no longer have
        // any input vectors to manipulate. Add the current result (rvi)
        // to the total set of results (rvvvi).
        rvvi.push_back(rvi);
        return;
    }

    // need an easy name for my vector-of-ints
    const Vi& mevi = *me;
    for(Vi::const_iterator it = mevi.begin(); it != mevi.end(); it++) {
        // final rvi will look like "a, b, c, ME, d, e, f"
        // At the moment, rvi already has "a, b, c"
        rvi.push_back(*it);  // add ME
        cart_product(rvvi, rvi, me+1, end); add "d, e, f"
        rvi.pop_back(); // clean ME off for next round
    }
}

See full code at Compiler Explorer.

Now, I’ll show you the recursive iterative version that I shamelessly stole from @John :

// Seems like you'd want a vector of iterators
// which iterate over your individual vector<int>s.
struct Digits {
    Vi::const_iterator begin;
    Vi::const_iterator end;
    Vi::const_iterator me;
};
typedef std::vector<Digits> Vd;
void cart_product(
    Vvi& out,  // final result
    Vvi& in)  // final result
 
{
    Vd vd;
    
    // Start all of the iterators at the beginning.
    for(Vvi::const_iterator it = in.begin();
        it != in.end();
        ++it) {
        Digits d = {(*it).begin(), (*it).end(), (*it).begin()};
        vd.push_back(d);
    }
    
    while(1) {
        // Construct your first product vector by pulling 
        // out the element of each vector via the iterator.
        Vi result;
        for(Vd::const_iterator it = vd.begin();
            it != vd.end();
            it++) {
            result.push_back(*(it->me));
        }
        out.push_back(result);

        // Increment the rightmost one, and repeat.

        // When you reach the end, reset that one to the beginning and
        // increment the next-to-last one. You can get the "next-to-last"
        // iterator by pulling it out of the neighboring element in your
        // vector of iterators.
        for(Vd::iterator it = vd.begin(); ; ) {
            // okay, I started at the left instead. sue me
            ++(it->me);
            if(it->me == it->end) {
                if(it+1 == vd.end()) {
                    // I'm the last digit, and I'm about to roll
                    return;
                } else {
                    // cascade
                    it->me = it->begin;
                    ++it;
                }
            } else {
                // normal
                break;
            }
        }
    }
}