I'll just comment on the mechanical points for now. I have not looked at the subclasses; so this is just about the class itself.
Armour: I like the exclusion of light armour in principal, it breaks the established rule though. For consistency with other classes you might want to add it.
Weapons: Fine
Tools: Armourer's tools are not part of the PHB I think. Maybe change it to Smith's tools? I personally think this should be a level 1 feature similar to Elegant Courtier from the Battle Master. Give one or two fitting suggestions, but leave it in the hands of the player.
Completely removing it, as tool proficiencies are generally handled by the background, is also fine I think.
Saves: Fine
Skills: I personally would switch Arcana for Religion and maybe add Medicine. Religion seems more along the lines of the (european) idea of a knight.
Equipment: Breastplates are way too expensive for level 1. Instead I would like to see a chain shirt. For weapons I would simply require two martial melee weapons or one weapon and a shield. Seems more natural to read. The equipment pack does not exist in the PHB. Again I would change it to something in the core rules and if I recall correctly all classes start with a choice of two packs here - diplomat and scholar seem fitting.
Mount: I would remove this or at least the difference in cost as bonus gold. This again just feels to different from the established starting equipment.
Ironside: Only usable in heavy armor. This makes medium armor even worse. I would extend it to medium armor. (This is extra weird, as you cannot start with heavy armor in this draft)
Challenge: Fine, maybe clarify what happens if the creature cannot attack you (dash?).
Leadership: This is very dependent on the order of combat. If six enemies can act after you and before your first companion, this might be completely useless. On the other hand if only friendlies act directly after you, this becomes super strong. I would change it to work until your next turn and increase the duration on certain levels. It's still very strong then if one has enough Vow points, but it's easier to track and less dependent on something that players have almost no influence on (initiative).
Extra Attack: The placement leaves 5 as a dead level. I really dislike this, as level 5 is a huge power boost for every other class in the book. Getting the third attack might annoy some fighters, but I think it is fair.
For Honour: This feels significantly worse than leadership for three times the cost. If the cost is adjusted (or leadership capped to once per rest) I like it.
General thoughts: I dislike the Vow resources management. It seems difficult to track (e.g. Courage) and lead to doing things just because you want the points (e.g. Mercy). I would remove the mechanical implications of this completely and make the points short or long rest dependent.
The Quest mechanic can probably be expanded to be a general thing for all characters with adequate rewards. As of now a Knight player might feel bad if their group is not eager to follow the knight's quest.