The ambition doesn’t matter. What gets you the good gear during a noble house war are two things.
One, missions where you are fighting alongside one noble house in a big battle against another noble house. If you play conservatively you can kill a lot of enemies (and take their stuff) without suffering bad casualties yourself: position yourself so that your guys are not exposed and let your allies get whacked on.
Two, once your band is mighty enough to take on noble troops straight up, fighting the town and militia patrols can yield quite good gear. Be careful not to take on too many at once as the regular troops can be tough.
Also, if you are just playing normally without daggers or flails, you are not going to get much in the way of armor. You will loot some nice weapons, but most of the good armor will get destroyed. If you want to get good armor, you need to use flails for headshots, and if you want both armor and helms you need to use daggers for puncture. Having 2 to 4 guys specializing in dagger or flail working as a unit to target high value enemies can yield some excellent games. Your whole front line doesn’t need to be dagger or flail dudes, but you do want at least some, to get the best armor.
There is one ambition which can help, which is the ambition to become Allied to a noble house. Once you meet that ambition you get some free armor, shields and maybe a helmet, all pretty good stuff, in the colors of that noble house.
Also, as you become friendlier to the noble house, you can also buy stuff more cheaply at their castles. If they have castles with high end weaponsmith or armorer, you can buy good gear. Once you get to that level, you may find yourself rolling in cash. (YMMV)