Can Claude or another tool write posts or answer the inbox?

Yes. Under Settings, then API keys, make a key and choose what it may do and for which brands. It can never do more than you can, and it can never manage people, billing or connections. Copy it when it appears, because it's shown only once.

Give it to an assistant that speaks MCP, like Claude, at the MCP address on that page, or use the API from another tool. Everything a key does goes through the same checks as a person, shows in the audit log under its name, and a brand that wants approval still keeps its posts waiting for someone. Revoke a key there and it stops working at once.