I know that the drag force is parallel to the direction of flow, thus equal to the unit vector of velocity. But if you consider a vehicle, when its angle of attack is below the velocity vector from a local vertical local horizontal reference frame, does this mean that the lift vector will be pointing downward too and not up?
Is it possible for the lift force unit vector...
Be careful in the wording, the lift unit vector has by definition a magnitude equal to 1, and by convention positive lift is up (or in certain contexts oriented to the suction side of the airfoil, this is equivalent unless flying inverted).
when its angle of attack is below the velocity vector from a local vertical local horizontal reference frame, does this mean that the lift vector will be pointing downward too and not up?
In simple terms, your question boils down to: Can lift be negative? Yes lift can be negative by adjusting the angle of attack to a negative value.
However the angle of attack is not between the direction of flight (velocity vector) and the wing chord, but between the airflow direction (relative wind) and chord:
The two notions are only equivalent in certain conditions, like when flying at null or constant vertical speed.
Negative lift is is visible on the curve showing the lift coefficient:
Lift and drag coefficients according to angle of attack (α), source (adapted)
On the left hand side, the orange curve is below x axis, and is negative. This occurs for values of α < -3°. So for values >= -3° lift is positive, and below lift is negative, meaning a small negative angle of attack is not sufficient to orient lift downward.
There are control surfaces naturally developing a negative lift value: Horizontal stabilizers (on many aircraft). They are used to prevent the nose to pitch down due to the center of gravity being forward of the center of lift.
Horizontal stabilizers developing a negative lift, source (adapted)
In the figure above, lift W is positive. It acts on a point which is aft of the center of gravity CG (by dw distance), this creates a pitching moment, the nose tends to pitch down. Horizontal stabilizers are used to produce a negative lift T. Given the longer arm of T (dt), this small negative lift can counter the pitching moment due to lift W. When the aircraft is trimmed this way the wing generates lift W equal to the sum of weight and T.