I was able to start the missions during King's Day, there's no problem on my end with starting Machine of War Quest on Kingsday!
I do know that he does say some different quotes on Kingsday like: "Baxter: LONG LIVE THE KING! Our wise and generous leader allows all commoners to celebrate Kingsday in the castle. What honour! Celebrate Kingsday on the roof of the castle. Come people, come! What a special day."
I recommend checking your quest log to see what's going on. As mentioned already in the comments, I know for burning the war equipment it says you need to wait some days before doing it again. The NPC in charge of the missions would say "Baxter: We already know their current plan! Come back in a few days. By then they might have changed their plans and I will have a mission for you." If you need to wait longer. You can complete the war plans part of the quest as many times as you like, independent of the repairing/burning parts.
Remember to say Mission, Orc War, War Plans to start the quest, and then check your quest log! I've tested this out on a character I've already done the quest on and one that I never did, works both ways no problem.
Source: Machinery of War Quest/Spoiler by TibiaFandom
Here are some pictures as proof of me starting the quest successfully on Kingsday in my world-