If I recycle a mech using Rumble Trigger and it's at Vilemaws - can I play the new unit at Vilemaws?

Yes, you can play the Mech from your trash onto Vilemaw's (the battlefield you just conquered) using Rumble, Hotheaded's trigger.

According to RiftJudge FAQ #9975, because Rumble's conquer effect creates a triggered ability that goes on the chain, you maintain control of the battlefield throughout the resolution of that chain (per Rule 187.4.c).

Here is the step-by-step sequence:

1. Trigger: You conquer Vilemaw's with Rumble, Hotheaded. This trigger is placed on the chain.
2. Chain State: The game enters a "closed" state. Per Rule 187.4.c, you cannot lose control of a battlefield while there is an item on the chain. Even though you may have just emptied Vilemaw's of units, you still control the location.
3. Resolution: When the Rumble, Hotheaded trigger resolves, you may recycle a friendly unit to play a Mech from your trash.
4. Finalization: As the Mech is played to the battlefield, it enters play. Because you maintained control of Vilemaw's throughout the entire chain resolution, it remains a valid location to play your new Mech.

This is a specific interaction confirmed by RiftJudge FAQ #9975, which explicitly notes that you do not lose control of a battlefield if you are in the process of resolving a chain item.