Each state does handle voting differently as the power is given to states.
But, for example, before the pandemic, you would go into the appropriate local governmental office in some states (such as NY) to request your ballot directly. It is worth noting that a driver's license is one of the acepted forms of identification in NY. Voting in NY is easier for people with driver's licenses, driver ID being ubiquitous. Proof of residence is also generally required when you register to vote.
It is also possible to fill out a Federal Post Card Application (FPCA), if you are, for example, in the military on active duty. The FPCA may be used by citizens who are overseas. The requirements of photo ID and proof of residence are waived for members of the military who use the FPCA.
Mail-in ballots that are sent out to registered voters suffer some problems.
1. Voters are unaware of the ballot.
2. Voters do not present themselves.
3. The ballots may be handled by various third parties.
For example, a person could fill out ballots for his parents, his parents would not be aware they had received ballots, and the local government would not know that ballots were submitted by someone other than his parents. That's just one simple example of how the system can be (and has been) exploited.