1040X is paper.
You need to research if the state(s) you are doing can be efiled. e.g. here is CA https://accountants-community.intuit.com/articles/1859786-ca-individual-amended-return-e-file-availa...
Here is a listing of piggyback states https://accountants-community.intuit.com/articles/1861800-ef-atlas-federal-state-ef-method-standalon...
My guess is that since you are worried about another REP, there is no extension. If you think the client will owe any of these entities, give them an address for each and tell them to send money.