Because the normal breaking of provinces occurs after the reaction window for winning or losing a conflict closes, the card described, Vengeful Oathkeeper, cannot be played in the situation given.
This question is fundamentally a question on timing, and for that, Appendix I: Timing and Gameplay located on pages 18 to 23 of the rules reference is the main reference point. Before getting into the framework steps, the two paragraphs for Reactions and Interrupts for framework effects should be noted. Essentially, Reactions occur in between two framework steps (between the framework step that created the effect and the immediately following framework step), while interrupts occur during the framework step. The section on Triggering Conditions (p. 16 & 17) is also advisable reading, but not strictly required in the context of this question (the only potentially relevant part is that simultaneously created triggering conditions are responded to in shared interrupt/reaction windows)
The relevant framework steps are in the Conflict Resolution table (3.2.x, page 19, detailed in pages 20-22). Specifically, 3.2.3 is the exact framework step that creates the framework effects of "win a conflict" and "lose a conflict" [of a given type]. It is at this step that a card with timing "when you lose a conflict" or "after you lose a conflict" would trigger. Later, framework step 3.2.5 breaks the target province if appropriate.
Suppose a hypothetical "when you lose a military conflict, draw 3 cards" interrupt effect exists and is triggered. Should the card described in the question, Vengeful Oathkeeper, be drawn here, it would trigger after any other interrupts occur but before the province (normally) breaks, as provinces do not get broken by framework effects in step 3.2.3.
Step 3.2.4 applies an "unopposed" bonus if the attacking player won in step 3.2.3 and the defending player did not have a participating character in step 3.2.3. The province still has not broken, and the window for responding to winning or losing a conflict has closed (meaning you can no longer use Reaction effect with those timings).
It is at step 3.2.5 that the framework event of "Break the attacked province" occurs based on whether or not the attacking player won by a large enough margin in 3.2.3. Assuming the province was indeed broken and the attacked province was The Art of War, its interrupt would trigger and the defending player would draw 3 cards. If Vengeful Oathkeeper was drawn in such a manner, it would NOT be played, as the current reaction window would be for "a province has been broken" and the window it would react in was closed in the previous step.
Note that should an attacker's interrupt or reaction to "You win a military conflict" perform the effect of breaking the province The Art of War during step 3.2.3, and as a result of the interrupt or reaction's effect of breaking The Art of War, Vengeful Oathkeeper was drawn, first the reaction window for the interrupt/reaction's effect of breaking the province would close, then the window for reactions to "You lose a military conflict" would be available again. In this hypothetical scenario, Vengeful Oathkeeper would react to the effect as the province was actually broken due to an interrupt or reaction to the framework effect of the attacker's "you win a military conflict" or the defender's "you lose a military conflict".