I had a child like this, and I can tell you that what NEVER worked and that was standing there in an increasingly awkward encounter trying to encourage, cajole, force, demand, she spoke up, whether to say Hello or Goodbye or 'Thank you for the gift/for having me.' while the other parent/relative got increasingly uncomfortable.
It was impossible to enforce, never resulted in a free, ungrudged, unmuttered greeting, and just made much more of a 'thing' than was necessary. It's not like the other person cared or even noticed until I made something of it. Why the hell would 'Sally's mum' care that 'Apples' daughter had terrible manners, especially if I breezed over it myself with 'oh wow, Applekid, isn't that lovely/what a lovely time you've had/thank you so much, she'll have so much fun with that!'
Lots have said the same as me, I see - IncompleteSenten said it really well "Saying hello for them and using lots of "we" sentences eg we're so glad we bumped into you, and things like "me and daughters name were saying (insert something nice about them) the other day." Modelling conversation without expectations.
Applekid is adult now and while she'll never be as 'gushy' as I am she can perfectly well manage nice, polite interactions, so honestly, it will come in its own time if you shelve it completely.