Which algorithm should I use to extract an array from a multidimensional array?

One relatively simple way is to approach this task as a recursive mapping of K-tuples of dimension N to K-tuples of dimension M < N. Since this is language-agnostic, the following facilities need to be available in your target language:

1. Given an instance of an N-dim array, find N
2. Given an instance of an N-dim array and a k < N, find the number of items in dimension k
3. Given an instance of an N-dim array A and a vector of N integers I, get or set the item at a position defined by indexes in the vector. For example, if the vector is I = {2, 3, 4}, there needs to be an operation for obtaining A[2, 3, 4] from the 3-D array A and a vector I.
4. Given a vector D of size M defining dimensions, there needs to be an operation for constructing an array of M dimensions, each dimension taking the size of the corresponding item in the vector. For example, if the vector D = {3, 4}, there needs to be an operation that creates a 2-D array with dimensions 3 and 4.

Preprocess the request by constructing vectors F and D as follows:

• For each "all" item at position i of the request, F[i] = -1 and D[i] = size(dimension[i])
• For each numeric item k at position i of the request, F[i] = k and D[i] = 0.
• Create an array for the result by taking all non-fixed indexes and passing them to the array constructor (prerequisite number four above).

Now your recursive procedure should look relatively straightforward:

void RecursiveCopy(Vector F, Vector D, Array S, Vector SI, int sp, Array T, Vector TI, int tp) {
    if (pos != TS.size) {
        if (F[sp] != -1) {
            // This is an "all" dimension
            for (int i = 0 ; i != D[sp] ; i++) {
                SI[sp] = i;
                TI[tp] = i;
                RecursiveCopy(F, D, S, SI, sp+1, T, TI, tp+1);
            }
        } else {
            // This is a fixed dimension
            SI[sp] = F[sp];
            RecursiveCopy(F, D, S, SI, sp+1, T, TI, tp);
        }
    } else {
        // Read from the source at indexes defined by vector SI
        var value = S.get(SI);
        // Write to the destination at indexes defined by vector TI
        T.setValue(TI, value); // Prerequisite 3
    }
}

Your getSubarray would look like this:

Array getSubarray(Array S, Vector<string> request) {
    Vector F = Vector[S.Size]; // The number of dimensions in A; prerequisite 1
    Vector D = Vector[S.Size]; // The number of dimensions in A
    Assert(request.Size == S.Size); // Request must have N items
    int k = 0;
    Vector resDim;
    for (int i = 0 ; i != req.Size ; i++) {
        if (req[i] == "all") {
            k++;
            F[i] = -1;
            D[i] = A.dimensionOf(i); // Prerequisite 2
            resDim.Add(D[i]);
        } else {
            F[i] = ParseInteger(req[i]);
            D[i] = -1;
        }
    }
    Array T = Array(resDim); // Prerequisite #4
    Vector SI = Vector[S.Size];
    Vector TI = Vector[k];
    RecursiveCopy(F, D, S, SI, 0, T, TI, 0);
    return T;
}