This will be an interesting question. Are all those IGs presidential appointments (I'm guessing not, see below)? Some IGs are appointed by agency heads, and in theory, the power of removal rests with the appointing agency head, not the President.
The courts have been very careful to never come up with a really firm rule on whether or not (or, if so, how) the President's power of removal may be limited.
My guess is Trump's lawyers had him make this move specifically to test this proposition, and I'm guessing at least some of these IGs are agency appointees, so that the courts will be forced to examine that issue, as well. My guess is that a large number of Trump's early moves will be intended to create court cases that help test and refine the so-called "unitary executive" theory, and I think these firings represent a clear example of that gameplan.