This is such a "trope" in science fiction. From an episode of X-Files where these vampires were all living in the same trailer park. To the movie Blazing Saddles where everybody in Rock Ridge was a member of the Johnson family.
Once you get even a few adults in, the rest goes pretty smoothly. Say you get five or six in some local town government bureaucracy. It all depends on the first few people getting in and getting a foothold in local city government. They can fake up required records such as birth certificates.
The first few may need to be gotten in through some kind of identity theft. Supposing you are nice aliens. Maybe you must follow around a lot of people and wait for somebody in a usable job in City Hall to die of natural causes. Then you sneak up on the corpse and BLIP! You have ID. If you are less nice maybe you have to hurry it along. If you are really horribly not nice, maybe you also have a snack.
Once the local community gets big enough, you can even have things such as the mayor, the police chief, and the town doctor being your pals.
Example: New subdivision goes up. It's operated by a "front" corporation, incorporated in the town by the original five people in City Hall. Every house is sold to your pals. This gives several thousand people a steady and stable home base from which to operate. They can send people out to nearly everywhere. Now that they have a home to come back to, and a home address to put on forms and things, they can do nearly anything. Go to university, run for public office, open businesses. The fact that their paperwork starts and stops with one particular City Hall is easy to obscure and hard to untangle.
Example: New trailer park gets built at the edge of town. All the permanent residents are your pals. Now they have an easy to defend origin story. They can then hop to other communities in other cities and openly integrate into society. Jim-Bob in the trailer park gets a job and becomes James, the accountant, with a semi-detached bungalow in Terracotta, Texas.
Example: Whatever the appropriate buzzwords are for making an immigrant look attractive, one of your pals fills in the form with that. And when he shows up, a local, also one of your pals, is there to sponsor him and support him. And he takes his place as a proper and legal immigrant, officially from Lower Slobovia,but actually one of your pals. The pals in City Hall make sure all the paperwork is properly approved, stamped, signed, and filed.
Once there is even a smallish community, they can start having a lot of babies. Then to avoid drawing attention as "the most baby-having town in America" they can start moving to other communities and set up the whole process again. And each new baby is duly and correctly registered as a new citizen of the country.
Edit to respond to the comment from Alexander:
Getting by this depends on how nice the aliens are. But basically, it requires burning down one records building, and then filing for a bunch of delayed birth certificates to make up for it. With a little jim-jam on dates, you could easily get several hundred people in this way. The "how nice are your aliens?" question revolves around how much damage, and how many people are injured or killed, in the fire? Possibly, if they are really nasty aliens, every person killed becomes one more stolen identity. Imagine that the fire takes place on "take your kid to work day" so that single-moms with one kid become family package deals.
The real trick is to not pull this on anybody who has fingerprints on file. If there are prints on file it could be awkward if the replacement is ever picked up by the FBI for some reason.