Variable for who filled out the census form?

Is there a variable that tells you who filled out the census form (e.g. enumerator, neighbor, head of household, etc)?

There is no such variable available in the Census samples. The closest variable is RESPMODE, but this is only available for the ACS. You can find information about the possible response modes by decade on the Census History website. Prior to 1960 all enumeration was done in person.