There have been some proposals previously to add Curve25519 and Curve448 to the Web Cryptography API, e.g. [Proposal] Add curve 25519 and cruve 448 to the web cryptography api and [Proposal] WebCrypto Argon2, Curve 448 / 25519 / secp256k1, ChaCha20-Poly1305, but there has been no draft spec yet.
I reached out to the browser vendors and got some positive responses, so I went ahead and wrote a draft spec, here: Secure Curves in the Web Cryptography API.
There is also an explainer (based on a previous proposal by Qingsi Wang to include Curve25519 in WebCrypto).
Let me know if you have any questions or feedback