Is it possible to publish a deceased relative's unpublished work with them as the author?
It is possible for you to submit things to arXiv, even without your name on them, but do note that you need to be "endorsed" to submit to the site. That basically means that someone with a good submission history in the same subject category has to vouch for you by filling out a short form on the site.
Any professor or (probably) senior postdoc in the field that includes your relative's research should be able to serve as your endorser, if they are willing to put in the time to verify that the research is legit. Usually these people don't have spare time to spend on reviewing research from unknown sources, but maybe if you have a personal connection with one of them, you could ask. Or I guess you could try emailing someone at a local university, but honestly you're unlikely to get a response.
If you can't get an endorsement for arXiv, you can still submit the work to a journal that covers the appropriate research area. There's no endorsement requirement, affiliation requirement, or such to submit something to a journal. Figuring out which journal to submit to can be a little tricky, though, especially if you're not in the field. Again, in this case, it would be really useful to have a bit of help from someone who is.
Finally, if you find it too troublesome to submit to either arXiv or a journal, posting the work online in any form (on your personal website, in a document repository, etc.) is a perfectly decent way to expose it to anyone who wants to read it. Of course if you care about getting people to read it, that's again going to be quite difficult without having someone to vouch for you.
Something you could do is pop into the Physics SE chat room and see if anyone is willing to look over the paper. There's no guarantee that anyone will, and I certainly wouldn't expect that anyone there would be willing to endorse you to submit to arXiv, but you might get some initial pointers in the right direction, e.g. people could probably tell you what subfield of physics it's in. (Full disclosure: I'm a moderator on Physics SE, not that I think it really matters here.)
Your relative may have already started the review process - with a journal - they may already have the paper under revision. If so, then it could be worth checking with that editor who may consider helping out somehow.
Edit based on comments : an example of a postumously published paper can be found here: https://m.phys.org/news/2018-07-scientist-paper-earth-future-climate.html
Assuming the next of kin that own the work agree about his desire to publish, I believe his would be fine. You may want to note where the corresponding author would be listed that it was uploaded to arXiv posthumously.