So normally the police or hospital is called and people get a death certificate.
If someone disappears with no report, what happens? Is a death certificate ever issued?
In lieu of a death certificate, a person can be declared dead by a court with probate jurisdiction under circumstances outlined under state statutes. Usually this is a combination of circumstances that make the likelihood of death nearly certain and/or the passage of a certain period of time (five years in California) without a sign of the person after duly diligent efforts have been made to locate them.
In the meantime, a court can establish a conservatorship or receivership for the missing person's assets to preserve them.
Detailed statutory citations for California are found here.
A person going missing is serious. Many thousands of people disappear every year. In most cases they return or make contact with friends or family after a short while, however, a significant number are never heard from again.
It is a MYTH that you need to wait 24 hours before telling the police of a missing person. Alert the police as soon as possible.
The quicker the authorities start to look for them the more likely there will be a happy outcome.
All jurisdictions have rules on when a person can be presumed dead, that is, a death certificate is issued notwithstanding the absence of remains. In the united-states this is a matter for state law unless there is a reason for the Federal government to be involved (e.g. military personnel missing in combat).
In most, a court or the registrar can decide that a person is probably dead and issue a death certificate/grant probate.
People who are known to have been in the area where a tragedy occurred may be declared dead shortly after that tragedy even though no remains are found. For example, people on the Titanic who were not rescued by the Carpathia, people in the 9-11 attacks, the people on MH17 were all issued death certificates shortly after the respective tragedies.
Alternatively, at common law (codified by statute in some jurisdictions) a missing person may be declared dead if:
There are interesting legal implications of a person who has been declared dead coming back but they are outside the scope of this question. Please ask another question if you're interested.
If someone disappears with no report, what happens?
Is a death certificate ever issued?
As for the OP's comment "they just never get a certificate until 100 years later when someone steals their SSN"