I am the academic referred to in the following answer by Schiphol:
- She edited her preprints to use her correct name. In the new preprints she adds a header with the reference to the published version (in the APA style, which is common in her field and also only uses initials of first names).
- Posted them to the relevant preprint service (PsyArXiv, in her case). PsyArXiv has fields for two DOIs: a "preprint DOI" and a "peer-reviewed publication DOI". The latter points to the older published version.
- She now encourages other researchers to cite her using those preprints and their associated preprint-DOI.
The essence of that answer is correct, the following is an elaboration. The three main points I would make are as follows:
Sometimes the journal will agree. In most instances, journals will refuse to alter metadata associated with the paper, but this is not universally true, as noted in Joerg Heber's answer. Some journals do allow this: answers asserting that it cannot be done such as this one are factually incorrect.
You can use preprint servers to generate competing metadata. If journals refuse to change their metadata, you are permitted to release your own via a preprint server (in my case, via PsyArXiv). Journal publication agreements will often (not always) offer you scope to do so and you can ask people to share only the version that uses the correct information. What I did was edit the name on my author-accepted manuscripts and posted those.
ORCID and Google Scholar can help. I changed my name on both ORCID and google scholar. The latter in particular is useful because the searchability of paper relies heavily on GS in practice, and GS allows you to merge records (e.g., it allows you to merge the preprint version with the journal version and specify which version you consider to be the correct one). Note also that GS indexes personal websites if you post author accepted versions there too (see this question) which will also have some effect.
The above answers the question from a technical perspective.
In addition to the technical aspects, there are some social and practical hurdles to consider when doing this. Most transgender people understand the importance and sensitivity of this issue, as it pertains to our personal identity, mental health and in some instances physical safety. Unfortunately, most people will lack this knowledge and you are likely to encounter resistance. With this in mind, I would add the following suggestions based on my own experiences. I suspect you have considered these issues already (in my experience transgender people almost always do!) but on the off chance that you have not...
Discuss with your coauthors. I spoke with my coauthors about my intentions to check whether they had any concerns. What I found is that most people initially didn't understand why it mattered to me but when I explained to them the importance of the issue everybody was fine. Most of my coauthors offered to help me with recovering source code from old manuscripts etc.
Don't take it all on at once. I found it distressing to go through the process. I'm old enough to have a lot of papers to edit: there are about 100 papers published under my deadname. Doing the editing brought back a lot of traumatic memories that are psychologically associated with the papers (e.g., one paper is associated with a person who raped me: attempting to edit that one set off flashbacks). This may not be a concern in your case, but many transgender people have trauma histories and those may come into play here.
Be patient. What I have noticed as that as more papers have accrued under the correct name, various automated "profiles" have started to use my name correctly.
As a final point, not for the original poster directly, but in service of the deeper goal of making this process easier in the future: I found it easier to solve this problem when people took my privacy & safety concerns seriously, and did not treat this subject lightly.
In light of the fact that there are very real concerns that the transgender members of our academic community have expressed about our safety and personal well being in connection to this issue, it is important that we start pushing to improve these systems and place some pressure on journals to change their practices.