It depends on what you mean by “legitimate”. I take it to mean as in an effective and successful explanation of a phenomena, better than alternatives. That is more sound, valid and likely than alternatives.
This would apply to, say, the standard model theory of matter, the theory of evolution through natural selection, the genetic theory of inheritance and so on. These are all legitimate, whereas,say, the Lamarckian theory of evolution is not. By contrast, Newton’s theory of gravitational attraction is still legitimate but not as a complete explanation but, rather, as a special instance of Einstein’s General Theory of Relativity.
Macroeconomics as a distinct field within economics was launched by Keynes’ General Theory arguing that classical, marginalist, market economics was only a special case – legitimate within its constraints. The main Post-Keynesian position is that that and it’s successors such as neo-classical, Real Business Cycles and New Keynesian theories and so on are not just special rather than general but also fail to apply in the real world.
Well Modern Monetary Theory is, by design, an internally coherent and externally consistent systematic formulation and enhancement of some of the best discoveries in Post-Keynesian macroeconomics. It more effectively explains many real world financial phenomena such as how government finance actually works and what tools the government has to manage unemployment, inflation/deflation, productivity, investment, consumption, trade and so on.
If you agree in this scientific sense rather than just due your subjective opinion, politics and preferences, it is legitimate and more legitimate than its aforementioned conventional/mainstream/ orthodox competitors as well as Austrian and Marxist alternatives.
Do not confuse this, as other answers do, with it being a critique of Neoliberalism and/or of left or right political values. MMT does indeed undermine Neoliberalism (but you do not need MMT to do that), showing not only what are the myths that neoliberalism requires as a political ideology but also why these are myths and what the implications are of freeing us from these myths. Neoliberalism has long infected not just right but also left political views.
So debunking Neoliberalism is not a specific attack on the right, just on an empirically unsound ideology that they more commonly employ. Too many on the left are opposed to austerity but still drink the Neoliberal Kool-Aid without realising it. Whilst, as a result, MMT appears to be more popular amongst some left thinkers, MMT itself, as a legitimate theory is not bound to any political value sets. Given whatever value sets one has, MMT can better and more legitimately show what the real likely outcomes of proposed policies than alternative and less legitimate theories.