Summary:
Our App → Send Message to HubSpot → Will HubSpot receive the message (including attachments)?
HubSpot → Send Message to Our External App (via webhook events) → Will we be able to receive the message (including attachments)?
Hi @angelesjm! Welcome to the Community-- happy to have you here 
Integrating a third-party chat directly into HubSpot’s Conversations Inbox has limitations. While HubSpot offers the Conversations Inbox API (currently in beta) for some interactions, full integration isn’t natively supported.
Is the expectation that any messages sent in your custom chat would be synced between HubSpot and your custom app or are you trying to utilize HubSpot’s native chat? I’d suggest providing more context about your ideal set up, as that might give our community members more to work with. 