I want to play a rogue but I'm afraid our party won't have any healers. There would only be 3–5 of us and though I know someone who might play a healer, I don't know if they'll actually join the group.
Is it possible to get an NPC healer in a party, and if so how does that work?
First, I'm going to challenge a presumption you've made. You don't need a healer in D&D 5th edition. It's very nice to have one, but every character can spend hit dice to recover hit points. There is probably a question and answer on this site on the subject.
Next, I'm afraid the only answer to your question is "ask your DM." The only mechanism defined is that skilled hirelings cost 2 gold pieces per day. What skilled hirelings are available and how they're hired is up to your DM.
Some important notes about 5e.
If you have a 5-person party and nobody has taken any of those classes, it would be rare. It would also mean that you have a party filled with damage dealing classes. With none of those support classes, your fights are likely to be quick and bloody. Again, bring healing potions.
Is it possible to get an NPC healer in a party, and if so how does that work?
If this still becomes necessary, this is totally up to DM fiat. However, if your DM would like some guidance, I recommend looking at the Adventurer's League DM's Guide. That's some official Wizards guidance on paying for various healing and restoration services in 5E.
Although 5ed has less concrete need for a healer, they're still nice to have. Your party is 3-5, hopefully not including the DM, so at the low end, you'll probably need some help.
This is a pretty common problem to have in D&D, across all the editions. The common solution in my experience is to include an NPC party-member who just doesn't show much initiative and generally just does what's asked. The alternate (and official) solution is a faceless hireling.
If you use the hireling: they're paid a wage and you tell them what to do, generally a player is responsible for their actions.
If you use an NPC party member: they'd usually get an almost equal share of the party treasure but because they have no player they're almost entirely passive, they only act when asked to do so, largely because portraying them is too much trouble for the DM who is already busy portraying the rest of the world. If the party is in particularly dire straits, then the NPC might lend a hand with some defensive spells but usually, it requires a player to request of the GM that the healer do something or other. This frees the DM to focus on their job and lets any players who are otherwise idle (incapacitated/dead/held) have something to do that contributes to the action.
When the action gets around to the healer each round, the party will generally know what they want done and ask for it to happen ("Can the healer help my fighter please"). The DM nods and somebody, probably a player, rolls dice for their healing and crosses off the spell or whatever.
The character sheets for both kinds of healers are generally public knowledge and held by the players though if a DM is feeling tricky then they might maintain a secret one that includes the levels of assassin and the ring of undetectable alignment .
