Although the ideal is to get everyone vaccinated, there currently are not enough doses, so while they’re making more, we get to argue about who should receive the doses that already exist.  Different suggestions imply something about underlying ethical principles and intuitions.  It seems mostly uncontroversial that health care workers ought to be vaccinated first.  If you can remember back to when we used to go places on airplanes, recall the safety announcements: please make sure your oxygen mask is secure before helping others.  The logic here is impeccable, if slightly counter-intuitive.   Even if you feel strongly that you need to tend to the needs of others above your own, if you pass out from oxygen deprivation, you can’t help anyone, plus you’re dead too, so that’s the wrong answer.  Securing your own oxygen first is thus not merely self-interested, it’s also the necessary condition for your helping others.  This is allegorical for lots of things, but particularly on-point here: if we’re worried about a global pandemic, it’ll just make things even worse if the health care workers get sick.  If they get sick, who will take care of me?  So it makes good sense for them to vaccinated first.  And of course we wouldn’t have this problem if there were more doses available. So people who work in labs that study and create the virus seem like they ought to receive priority as well.

After that, it’s less obvious.   Some considerations pro and con for various candidates:
The Elderly
Pro: they’re more at risk of serious complications and death
Con: they’re also more at risk of dying of other things
The Young
Pro: they’re more likely to violate social distancing protocols and participate in spreader activity. 
Con: they’re less vulnerable to serious complications and death.
Note that the analysis of old-vs-young folds in on itself.  It’s not the 80-somethings who are going to bars, nail salons, gyms, frat parties, the mall.  So if the main concern is spreader activity, that’s an argument for vaccinating younger people, but if the main concern is harmful consequences of actually getting the virus, that looks more like an argument for vaccinating the elderly.

Front-Line Workers – no one has a precise definition for this, but it seems to be a way to categorize people like the grocery store workers and bus drivers.  The argument here is that we are all dependent of the continued functioning of things like supermarkets and transportation systems, so it’s in everyone’s best interest to make sure they’re healthy.  But how about:
Teachers – since the schools are closed, not only are students suffering from suboptimal education, but parents of school-age kids have had their work disrupted, and in many cases were obliged to stop working entirely.  And indeed, many heath care workers and people in the “front-line” category are parents of school-age children, so it’s in everyone’s best interests to reopen the schools as soon as possible.
The counter-argument to both of those is that there’s no precise and uncontroversial way to prioritize how important one job is relative to another. 

What about political leaders?  If politicians get sick, how will we ever manage as a nation? Maybe the sarcasm of that remark doesn’t translate into writing, but seriously, one argument in favor of vaccinating political leaders is that it might mitigate the sort of conspiracy-theory resistance to vaccination.  While I don’t think politicians deserve greater protection from the virus, there’s a consequentialist argument for them being vaccinated publicly, if it helps disabuse people of irrational fears.

Another dilemma arises from the suggestion that even though one is supposed to get two doses, maybe we could trade off instantly doubling the supply for mitigated effectiveness.  That’s a different sort of approach to thinking about who gets it.  That seems like a tradeoff that, in principle, we could evaluate empirically, but in reality will take more time than we have.  So proponents of either are gambling, to some extent.  If two people have headaches, and there is only one 500-mg Tylenol in the house, one way to go would be to have one person take it, reducing the total number of headaches by half, but the other way to go would to break it in half and give each person 250 mg.  That seems “more fair” in one sense, but would this result in two people with slightly-improved headaches, or two people who still have headaches?  If it’s the latter, that means the medicine has been wasted.  Again, before moral reasoning can be applied, we’d need some empirics.

So, it looks like the right order of priority is:
Health care workers
Lab scientists and workers who study and create the vaccine
Philosophers (because the rest of the dilemmas are still not obvious)