Is it appropriate to ask an editor to have double blind peer review even if journal does not offer such an option? The journal that I have chosen for submitting my paper doesn't use double-blind review
(double blind review  means that both the reviewer and author identities are concealed from the reviewers, and vice versa, throughout the review process).

Can I ask editors via email, or make a request for one? Is it difficult for them to organise the editing of the paper afterwards if it is approved or it is appropriate to do that?
With the submission management systems of most journals, it will be an incredibly obnoxious hassle for the editors to support a double-blind review if the journal's policy is normally single-blind review.

The reason is that, even if you remove names from the manuscript, most review systems present the reviewers with all of the key meta-data about the paper, including the author identities.  This information is tied to author accounts in the system, the paper production and publication process for accepted papers, and possibly also external systems such as ORCID.  This is configured right into the software, so the only method most editors would have for supporting double-blind would be to have the authors create pseudonymous accounts and then later work with the production staff to manually intervene and change all of the metadata (likely screwing up their records as well) before the paper actually appeared.  Thus, given the degree of hassle and problems involved, I would expect most editors to refuse without some sort of exceedingly unusual extenuating circumstance.

You can always ask politely, of course, but you should expect the answer to be no.