This PDF has a good list in Table 1.1 of statutes that grant the President wide lattitude in setting Tariffs. Tariffs are not covered under Article I, Section 8 as you quoted, that sentence applies to only internal interstate trade within the boundaries of the United States. The President has much more authority with external trade since that is a form of foreign policy. The more pertinent quote from Article I, Section 8 is a little further down:
To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes.
President Trump ordered a review of aluminum imports under the Trade Expansion Act of 1962 in April 2017, with the implication being that undercutting of the US Steel Industry would impact national security. The current action implies that this review is complete and he has decided to move forward in this interest.
There is also the Trade Act of 1974 that explicitly gave President Gerald Ford broad authority to "counteract injurious and unfair foreign trade practices." From Wikipedia:
Under Section 301, the President must determine whether the alleged practices are unjustifiable, unreasonable, or discriminatory and burden or restrict U.S. commerce. If the President determines that action is necessary, the law directs that all appropriate and feasible action within the President’s power should be taken to secure the elimination of the practice.
It's unclear from my reading of that Wikipedia article whether this act is still in force, it was initially set to sunset but was renewed all through the 80s and 90s. Parts of it were challenged in the 90s but the WTO rejected the challenge.
The President (or anyone else) can claim anything, and anyone with standing can challenge what they claim in court. If Trump intends to force the issue, I don't think he would also intend to ask for Congress' permission. But there may be enough will to challenge the action in which case the Supreme Court would probably intervene and determine if he has the authority he claims or whether other pertinent facts are correct (such as if the US steel industry is actually being undercut and if that affects national security). I doubt this will happen, since the order includes exemptions for Canada and Mexico and future exemptions are possible. This would allow the administration flexibility to address the concerns of anyone who may wish to challenge it who also has standing (Congress).
Historical Perspective
The first tariff was signed jointly by both the First Congress and President Washington (Tariff of 1789). In the early beginnings, the majority of income to the federal government was collected in the form of tariffs. Alexander Hamilton as Washington's Secretary of the Treasury attempted to levy excise taxes on local goods, which led some areas of the country to revolt in what came to be known as the Whiskey Rebellion. This wound up helping Thomas Jefferson win election over President John Adams in 1800 (after he narrowly lost the presidential election to Adams in 1796) since he campaigned heavily against the tax. Until the War of 1812 tariffs were the sole income, and this heavy reliance on taxes of foreign trade made it much more controversial than it is today.
In the run-up to the Civil War, the Whigs and the Democrats alternatively raised and lowered tariffs as they waned in and out of power (see Compromise Tariff, Black Tariff, Walker Tariff). The more industrious northern states exported more than they imported and wanted more protection, and the more agricultural southern states the reverse. But all of these actions went through Congress and were assented to by the President of the time (otherwise they would veto).
All the way up to the Smoot Hawley Tariff of 1930, legislation was drafted and adopted by Congress to set the different tariff rates and was generally one of the biggest economic campaign messages for the parties of the day. People representing areas with big industries were generally for more protection, and people representing areas that depended more on outside trade wanted less. During the Great Depression, Congress began to abdicate this responsibility with the Reciprocal Tariff Act, authorizing the Executive to negotiate on the behalf of the United States with the intention of lowering trade barriers with targeted nations (at the time mostly in Latin America). This "trade liberalization" continued after World War II as America was not only one of the few industrialized nations that was not ravaged by war, but they were also looking to use capitalism as a means of keeping communism from spreading.
The current laws the President is relying on is a continuation of this trend of granting the Executive branch more leeway in conducting foreign policy for Congress. But Congress does have the authority to provide a check and balance to this power as your question rightly alludes to. After Trump's announcement, his fellow Republican Jeff Flake has threatened to introduce legislation to nullify Trump's actions.
