I recently had a paper accepted where I included the ORCiD in the address field of the LaTeX code, as suggested in my comment to Carlo's answer. The publisher did not ask me for any form-based entry as part of the publication process.

The publisher (Springer) did automatically recognise the ORCiD identifier from this and included it in the metadata. IIRC the paper was automatically added to my ORCiD record by Crossref upon publication as a result.

This may not be representative of other publishers, but I do conclude that it is a good idea to include an ORCiD in the address field, and that there is evidence that this is picked up as part of the publication process.