At this time, custom participant data is not supported in Web Messaging. We do plan to add support for this, but right now there is no clear timeline of when to expect this.
As an alternative, you could consider using a bot flow via Architect to help collect this customer data as part of the web messaging conversation.
I suppose the deployment key could be surfaced via a Data Action within the flow. However, there is no native function making it available at this time.