I don't think any church would get into trouble with the government for promoting vaccination..That's actually the kind of thing that the government would smile upon.
Trying to avoid politics in this as much as possible...The more subtle issue is if vaccination would be considered "political" within a church. Some of the more evangelical branches of Protestantism seem to have firmly aligned themselves on a certain side of the politcal aisle, even embracing a certain politician, that to put it lightly, seems to stand for almost everything that Christians should be against. And it is no secret that this certain side of the political aisle is far less enthusiastic about promoting vaccinations than the other. So, in this context, the partisan divide might bleed into the internal workings of the church, so much so that fully embracing and encouraging vaccination might be seen also as a betrayal by joining the "other side" of the political aisle.