AFAIK the president has the power to borrow money, which is why the debt ceiling is an issue every time we get close to it because Congress orders the government to spend more than its making so the President has to borrow. The only illegal part would be the sending checks part, since that's spending money that hasn't been appropriated and isn't covered by an exemption.

That said, there's a civil war-era law that could theoretically be used to justify paying the troops without an appropriation. It's only intended to apply to feeding troops (and their horses) but I can imagine that being argued over in the Supreme Court.

It would be a pretty big power grab by the presidency to take direct control of that process away from Congress but its not outside the realm of possibility given the current trajectory of the government and the stated goal of the right to concentrate more and more power in the executive.