[Vault POC] PegIn flow

[jrwbabylonlab]

Implement the full PegIn flow by integrating all the components already developed in the preceding tickets. This task focuses on stitching together client methods, wallet interactions, API calls, and service layers into a working end-to-end PegIn process for the POC.

Scope:
1. Orchestrate all previously implemented methods (Vault Controller client methods, Vault Provider API, BTC/ETH transaction handling, wallet connections).
2. Ensure the depositor can:
• Select a Vault Provider
• Submit a PegIn request (ETH tx)
• Sign required BTC and ETH transactions via connected wallets
• Receive updates as the PegIn moves through its lifecycle
3. Integrate with local storage handling for intermediate states (e.g., BTC confirmations).

Notes:
• This is not about implementing new low-level methods (those are covered by other tickets), but about wiring them together into a complete user journey.
• UI can be minimal, but must reflect the main states to demonstrate flow completion.
• Borrow or mint vaultBTC is not part of this ticket.

Requires:

• [Vault POC] Implement Client Methods for Vault Controller (Ethereum Queries & Tx Submission) #259
• [Vault POC] ETH txs #272
• [Vault POC] BTC transactions #269
• [Valut POC] UI skeleton #264