Can my opponent react to me using Vanguard Armory?

Yes, your opponent can react to you using Vanguard Armory.

According to the game's timing rules, activated abilities (abilities that require a cost to use, such as exhausting/tapping the card) use the chain. When you exhaust Vanguard Armory to use its ability, that effect is placed on the chain as a pending action.

Before the ability resolves and the three Recruit tokens are played, your opponent receives a window of priority to play Reactions or other fast-speed effects. If they have a way to counter the ability or remove the Vanguard Armory in a way that interrupts the effect, they may do so while it is still on the chain.