Actually, according to the Reciprocal Tariff Act of 1934, the President has a large amount of power in dealing with tariffs....
https://en.wikipedia.org/wiki/Reciprocal_Tariff_Act
Yes, but only in the area of REDUCING tariifs - not in raising them.
"The Act served as an institutional reform intended to authorize the president to negotiate with foreign nations to reduce tariffs in return for reciprocal reductions in tariffs in the United States."
"Another key feature of the RTAA was the fact that if Congress wanted to repeal a tariff reduction, it would take a two-thirds supermajority."
And even if the president negotiated a change in tariffs, it had to be approved by Congress - a reduction in tariffs only required a majority vote, but an increase required a 2/3 majority.