Does Vanguard Armory spawn recruits on the turn it is played?

Yes, Vanguard Armory can spawn Recruit tokens on the turn it is played, provided you have the necessary resources and the game state allows it.

Analysis of Timing and Mechanics

• Playing the Card: Vanguard Armory is a Gear card. Like all Gear, it enters play Ready (Rule 149.1).
• Activating the Ability: Because the ability requires exhausting the card ([T]), it is an activated ability. Activated abilities can be used during your Main Phase in an Open State (Rule 381). Since the card enters play Ready, you can exhaust it to activate its ability immediately after it enters play, provided you have priority and the state is Open.
• Chain Interaction: When you activate the ability, it is placed on the chain as a pending item (Rule 377.3). Your opponent has an opportunity to respond with [Reaction] cards before the ability resolves (FAQ #8476).
• Resolution: If the ability resolves, you play three 1 [M] Recruit unit tokens. These tokens must be played to valid locations—your Base or a Battlefield you already control (FAQ #8577, #8845).

Summary

You can use the ability on the same turn you play the card because Vanguard Armory enters play Ready, and there is no rule preventing you from activating a Gear's ability on the turn it enters play, provided you are in an Open State during your Main Phase.

Note: I could not fully verify this ruling against the retrieved rules.