That's actually one of the things that has kept me from using Claude Code web (that, and I often need a Chrome browser for the agent). But they must be working on it.
I saw an MCP I've set up on claude.ai show up in my local Claude Code MCP list the other day, it seems inevitable that there will be skills integration across environments as well at some point.
In working on Rodney I found out that the Claude Code for web environment has a Chrome browser installed already. It's a shame you can't see its output directly - even if it takes a screenshot there's no easy way to view it other than having it commit and push that to a branch in GitHub.