Both commenters have pointed you in the right direction. Let me lay a broader foundation.
Congress's authority over the creation and use of substances or products is greatest under the Commerce Clause (Article 1, Section 8, cl. 3), which provides that Congress shall have the power:
to regulate commerce with foreign nations, and among the several states, and with the Indian tribes
Additionally, Congress has the supplemental power to "make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers (Article 1, Section 8, cl. 18). Together, these amounted, at the time of their drafting, to a major improvement over the Articles of Confederation, one of the major defects of which was its silence towards the Union's joint commercial interests. James Madison wrote about this very subject in [Federalist No. 42](http://federali.st/42#p9(http://federali.st/42#p9).
Just how much of an improvement they amounted to, though, has been a continuing issue in American law and history. The Commerce Clause more than others has been a political football since the time of its drafting, and it's easy to see why. Expand the Commerce Clause, and the federal government can reach farther into state economic affairs; contract it, and its reach subsides. One of the young nation's very first political fights, in fact, was a constitutional one over whether these two clauses authorized Congress to charter a national bank. (The Supreme Court held they did; the bank was eventually abolished anyways, under President Jackson.)
There is some authority showing that Congress intended the Commerce Clause to sweep very broadly indeed: Professor Amar at Yale has written extensively on the matter. (One such book is on the book list.) However, I'm not sure it has ever been interpreted broadly enough to encompass Prohibition. And regardless, before the 1930s, or the 1950s at the latest, broad interpretations of the Commerce Clause were hamstrung by other constitutional limitations.
Without regard to later legal developments -- and there have been many, which the other commenters have pointed you towards -- this means that in 1920, when the Eighteenth Amendment went into effect, lawmakers would likely have entertained real doubts about whether they could constitutionally ban the importation, creation, and sale of alcohol. They would have felt constrained by:
the limits of the Commerce Clause,
the limits of due process,
the Tenth Amendment, or,
all of the above.
Because I'm not a historian of the era in particular, I can't tell you that this is the reason Congress settled on a constitutional amendment. But I can tell you that prevailing legal thought at the time meant that an amendment was a more obvious course than an act of Congress.
I just wanted to echo the comment of AmesCG. The 10th amendment reserves all powers not granted to the federal government for the states. States therefore generally set out laws governing the conduct, health, morality, etc. of their residents, including the drinking age. Since the 10th amendment predated prohibition, they needed another amendment to affect a power traditionally reserved for the states.
Today, the federal government has broad spending powers (how it spends money it collects in taxes). The federal government controls the drinking age through highway spending in the National Minimum Drinking Age Act. Under that act, if a state does not enforce a 21 drinking age, Congress can withhold a percentage of highway funding. This funding is very important for states (essentially free money for roads) so states enforce a 21 year drinking age.