The 14th amendment does not address this issue and is not about conducting the census. The constitutional mandate is Article 1 Section 2:
In Article 1, Section 2, the Constitution includes the phrase:
[An] Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct.
Congress first met in 1789, and the first national census was held in 1790.
There was actually some debate about whether, how, and on what timetable a census should have been held. In early 1790, several members of Congress argued against a census prior to the next election. Some in the Congress, who advocated an immediate census, noted that those who did not want one were the people from states which were generally regarded as being over-represented in the Congress based on the initial figures provided for in the Constitution. Others were concerned about the questions to be asked in the census, while others felt that more questions should be asked to get a better picture of the citizenry.
For example, on February 2, 1790, Samuel Livermore of New Hampshire lamented that the proposed question about profession would be hard for his constituents to answer, since some had three or four professions, depending on the season. Connecticut Representative Theodore Sedgwick, on the same day, wondered why the questions were not extended further — "The state of society could be ascertained, perhaps, in some degree, by observing [the] proportions."
The final bill, Statute 2 of March 1, 1790, provided that census marshals and assistants be appointed. The marshals were directed to:
cause the number of the inhabitants within their respective districts to be taken; omitting in such enumeration Indians not taxed, and distinguishing free persons, including those bound to service for a term of years, from all others; distinguishing also the sexes and colours of free persons, and the free males of sixteen years and upwards from those under that age.