The president is head of the executive branch. The executive branch is the branch that executes the legislation passed by the legislative branch (hence the names of the branches). An executive order is simply a directive towards the executive branch, or some subset, regarding this execution. Only the executive branch is directly subject to the authority of the executive order itself.
However, the executive branch in turn has authority over the country. For instance, the National Emergency Economic Powers Act of 1977 allows the president to block trade with foreign powers. If we were dealing with a physical commodity, Trump's executive order would, explicitly or implicitly, direct Customs to prohibit the import of that commodity, with Customs being part of the executive branch and therefore under Trump's authority. If you want to import the commodity, Customs is going to stop you, and as someone entering the country, you're subject to Customs' authority.
States are free to ignore the executive in that they don't have to take any affirmative steps to enforce it. If a Customs official is standing next to a state cop, and says "That person just smuggled in some contraband! Arrest them!", the state cop is free to shrug and say "Arrest them yourself". But Customs is still going to be inspecting imports and blocking the banned commodity. It's similar to the Sanctuary City issue: when an illegal immigrant is arrested, the state is free to release them rather than transferring them to ICE custody, but ICE is free to stand outside the courthouse and arrest them as soon as they leave.
If a state or company ignores the ban in the sense of actively importing the contraband, then it's going to face criminal penalties.
In this case, there isn't a physical commodity being imported, so the question of just what federal agencies will be involved in enforcing it is more complicated, but the basic principle remains.