From a different angle - why a delta wing in the first place, and what reasons are there not to have an HS?
Delta wings are generally used for supersonic and high subsonic flight, because they have advantages regarding supersonic shock waves and reducing drag.
In that regime, the fewer the shock-producing surfaces and protrusions, the better. So it is advantageous to do away with a conventional HS, replacing it (as td and madscientist say) with elevons on the single wing. See also: Me 163, although that is a swept rather than a delta wing. but still lacks an HS.
Another point is that an HS can get caught in the shockwave off the wing, and end up with controllability problems at high speeds. While also being subject to a deep stall at slower speeds (HS being deprived of airflow by the "shadow" of the main wing at high AoA, and thus losing any effectiveness).
Additionally, a feature of long-deltas (e.g. Concorde) is that at lowish speeds, they can produce vortex lift at high AoA (tornados on top of the wings) - eliminating the need for high-lift devices (slats and flaps), and thus providing space on the wing for elevons.
Some deltas have canards to take the place of a HS, providing nose-up-lift at low speeds, and also handle the shifts in center-of-lift produced by shock-waves.
There is also the case of the Rogallo Delta often used for hang-gliders - where control is via the flexibility (warping) of the whole wing.
So it becomes a chicken-and-egg problem - which configuration or set of dependent features best suits the overall design goal of the particular aircraft? Does an HS cause more problems than it solves - or does eliminating an HS cause more problems than it solves?