Consider some collection of vectors satisfying the requirements. If we apply any regular linear transformation, we get another collection satisfying the requirements. There are $(q^n-1)\cdots(q^n-q^{n-1}) = q^{n^2} (1-q^{-n}) \cdots (1-q^{-1}) \geq c_q q^{n^2}$ of these. However, we might get the same collection. Let us call two collections equivalent if there's a regular linear transformation mapping one to the other. This is an equivalence relation, so if we upper bound the size of an equivalence class, we will get a lower bound on the number of collections.
How many regular linear transformations can take a collection into itself? Choose some arbitrary $k$ vectors from the collection. These are mapped into some other $k$ vectors from the collections (less than $l^k$ possibilities). How many regular matrices map the former to the latter? Assuming that the first $k$ vectors where $e_1,\ldots,e_k$, the first $k$ rows of the matrix are forced, and for the rest we have $(q^n-q^k)\cdots(q^n-q^{n-1}) \leq q^{n(n-k)}$ possibilities. Thus an equivalence classes consists of at most $q^{n(n-k)} l^k$ regular matrices, and the resulting lower bound on the number of collections is $\frac{c_q q^{n^2}}{q^{n(n-k)} l^k} = c_q \left(\frac{q^n}{l}\right)^k.$