Yes, but it may not be worth it.
In principle it is absolutely possible to use thrust for attitude control of a vehicle. To cite a few examples, the F-35, X-15, and Harrier all used ducted jet exhaust or high-pressure bleed air for attitude control during low dynamic pressure phases of flight.
You correctly point out in the question that the thrust would have to be at the nose (not normally where you want your engine) and inclined so the thrust line is 90 degrees downward.
So another way to ask this question would be "can you replace a canard on a delta-winged aircraft with a vertically thrusting engine"? In principle, yes. Is it a good idea? Probably not. But how good or bad of an idea it is depends on several factors. Some of the challenges have already been mentioned so I won't repeat them here, but other things to consider are:
1) How far the nose is from the C.G. Your concept makes more sense if you have a long nose on the airplane (think XB-70). For a pure flying wing you would need a lot more thrust to achieve the same pitching moment since the moment arm from the thrust location to the C.G. would be as small or smaller than the moment arm from the flap to the C.G.
2) The type of propulsion system used Tailless aircraft are very sensitive to C.G. location, so if you add a lot of weight to the nose that will make the aircraft difficult to balance. For that reason, it's more advantageous to use lightweight secondary propulsion systems (ducted exhaust from a turbine). Putting a large electric fan or secondary turbine in the nose will add weight in a difficult place to balance. There is a trade here with the benefit you get from a longer nose - you'd have to actually work it out for your design to see if there's an option that makes sense.
3) Pitching moment variation with speed
Dynamic stability (include gust response, pilot handling characteristics, etc.) is going to be a challenge for this type of plane in general. But it's a solvable problem; whether its solvable with a computer in the loop or not depends on some of the other details of the design, but flying wings, canard aircraft, and delta wings have all been built which require little or no stability augmentation.
There is one big stability challenge that this concept brings with it, which is that the effective pitch-up moment you get from the thruster will change relative to the flap pitch-down moment as speed varies.
Typically you'd calculate the trim condition of the vehicle in terms of key dimensionless parameters like $C_{L_\mathrm{tail}}$ and $C_{m_\mathrm{wing}}$, and when the aircraft is in a trim state the variations in these parameters with speed are very small; so small changes in speed won't effect the trim condition of the aircraft.
If you're using a thruster to generate your pitching moment the force of the thruster is dependent almost entirely on the jet velocity of the thruster, and almost totally independent of the freestream (especially in this case where it's oriented 90 degrees). This means that, as your speed changes, you'd need to constantly be modulating the thruster input to keep the right pitch trim. Since unlike other reaction control aircraft your thruster will always be on, keeping it at the right power setting will at best add significantly to the pilot workload during the highest workload phase of flight; it's very likely you'd need a computer in the loop to make this concept work.
Bearing in mind the weight/complexity, c.g., safety, and control issues this brings up its worth thinking hard about what this buys you; adding a canard or a horizontal tail are much simpler ways of accomplishing the same thing.