TL,DR: it all boils down to a single, poorly-verified self-report. But since the whole claim is a report about one person's feelings and plan, there's little substance in the first place.
The story has been reported in the French press recently, but it isn't new. A recent report is ‘Interview : Lilly, pionnière robosexuelle, “Que du bonheur”’ (‘Lilly, robosexual pioneer: “pure bliss”’), in Maïa Mazaurette's blog, dated 20 December 2016. I would normally consider this a reliable source as Maïa Mazaurette also writes for the French reference newspaper Le Monde. But as of today, six days after the blog post, it has been pulled. The entry is still visible in Google's cache, with a photo. It does include the claim of “fiançailles” (engagement) to the robot, which was built by the woman in question, and a desire . So I would treat the existence of someone who claims to be a woman “engaged” to a robot she built as proven, but the validity of that claim is subject to caution. The blog entry may have been pulled because the journalist realized that the entry was a fake, or because the person desired to reclaim some privacy.
A blog entry dated the previous day is still online, announcing the interview for the next day. It claims that the woman “created [her own robot] in 3D”, “defines herself as robosexual” and “waits for a country to legalize human-robot marriage”.
The interview cites an article on futureofsex.net dated 1 December as well as the now defunct @Lillyinmoovator Twitter account. The futureofsex.net article, in turn, only cites the Twitter account and an interview with Lilly.
A Facebook event announced the engagement of Inmoovator and Lilly on 30 October 2016. I didn't find any trace of any ceremony.
Note that unlike marriage, “engagement” (fiançailles) is not a legal framework, only a self-declaration. (Marriage is defined by the Code civil, which does not mention fiançailles at all. An older law mentioned “promise of marriage or engagement” (promesse de mariage ou fiançailles), but only in the context of paternity declarations, and that article was repelled in 1993.)
None of these reports mention any concrete plan for a legal framework for a human-robot marriage, only a hope that this would come up in the future.
The robot would presumably be based on the InMoov design, which is designed to be built by an individual with affordable equipment. (There are numerous reports in Le Monde, e.g. “30 years after its invention, 3D printing technology makes rapid progress”). This of course doesn't say anything about the claim we're interested in other than ruling out one way it could be implausible.
Overall all the reports seem to stem from a now-dead Twitter account (@Lillyinmoovator) and some online interviews with the presumed holder of the account, plus a few photos of a woman with an InMoov robot, all of which first appeared on said Twitter account. The first secondary source was the futureofsex.net interview on 1 December in English, then there was the 20 December interview in French, then a lot of tertiary sources around the world picked it up.