If a player plays Vanguard Armory and taps it, can I react to either of those?

Yes, you can react to the activation of Vanguard Armory, but you cannot react to the act of playing the Vanguard Armory card itself.

Here is the breakdown of the timing:

1. Playing Vanguard Armory (The Card)

When a player plays a permanent (such as the gear Vanguard Armory) from their hand, it is placed on the chain and resolves immediately. According to RiftJudge FAQ #7617 and Rule 359, playing a permanent does not pass priority on the chain, so there is no window for your opponent to react to the card entering play.

2. Tapping Vanguard Armory (The Ability)

Once the Vanguard Armory is in play, its ability—which requires exhausting (tapping) the card—is an Activated Ability.

According to RiftJudge FAQ #8476:

• Activated abilities use the chain.
• When the controller exhausts the Vanguard Armory to use its ability, that effect is placed on the chain as a pending item.
• Once the ability is on the chain, the state becomes "Closed," granting you an immediate window of priority to play Reactions or other fast-speed effects to interact with that ability before it resolves (e.g., to remove the Vanguard Armory or counter the ability if you have a legal reaction).

Summary of the sequence:

1. Player A plays Vanguard Armory: It enters play immediately; no reaction window.
2. Player A activates the ability (taps): The ability goes on the chain.
3. Priority passes to you: You may now play a Reaction to the ability currently on the chain.