Politics. If we stipulate that secession was illegal, that doesn't mean that particular members of Congress are going to pull the trigger to do something about it. And pulling the trigger on a civil war was a tough decision for a lame duck Congress and a lame duck president.And if any of that was an act of war, Congress (who was still in session during all that time, I believe) should have declared war. That is the constitutional duty of Congress, not the President. No one did anything to stop the actions in the Southern states. Why not?
Modern politics shows us that even when faced when crisis situations (name one based on your political leaning), Congress will be slow to act, if they act at all. Secession was a shock trauma that basically waited for Lincoln and the 37th Congress to address.
What Congress did try to do was, how do I say it... seek a conciliation with the secessionists by, for example, approving the so-called the Corwin Amendment on March 2, 1861. It would shield "domestic institutions" of the states (which in 1861 included slavery) from the constitutional amendment process and from abolition or interference by Congress. It was submitted it to the state legislatures for ratification.
This attempt, as well as probable others, simply did not move the needle for the secessionists. I think some Congressman (and perhaps Lincoln) believed that secession was a bluff to force a political advantage, or that Unionists sentiment would come to the fore and end secession; and therefore they could wait out on secessionist sentiment. But those were all pipe dreams. In the end there was no easy way out. And then the war came.