If I use Irresistable Faefolk to target and move a unit, can I react to my own Faefolk action before it resolves?

Yes, you can play a [Reaction] to the chain after you have placed the triggered ability of Irresistible Faefolk onto it, but before that ability resolves.

The Reasoning

According to the Chain Rules and Section 327 (referenced in the context), triggered abilities such as "When I move to a battlefield..." behave like activated abilities and are placed on the chain.

When you move Irresistible Faefolk to a battlefield, its ability triggers. You (as the controller) choose to place this ability on the chain. Once this ability is on the chain, the game enters a "Closed State" (defined in Rule 331.1 as occurring when a chain exists). During a Closed State, players with priority have the opportunity to play [Reaction] cards to add to that chain before it resolves.

Sequence of Events

1. Move: You move Irresistible Faefolk to a battlefield.
2. Trigger: The "When I move to a battlefield..." ability triggers. You place it on the chain.
3. Chain State (Closed): The chain now contains the Irresistible Faefolk trigger.
4. Reaction: You (or your opponent) have the opportunity to play a [Reaction] card, adding it to the top of the chain.
5. Resolution: The chain resolves in Last-In, First-Out (LIFO) order. Any [Reaction] you played will resolve first, followed by the Irresistible Faefolk trigger.

As noted in RiftJudge FAQ #11094 and #10923, the movement or removal of the source of a trigger (like the Faefolk) after it has been placed on the chain does not invalidate the trigger. It remains on the chain and will resolve successfully, referencing the location where it originally arrived.