There isn't really a ubiquitous reading of the principle of sufficient reason (PSR), so I think it's worth going over some distinctions and different formulations.
There are at least two major formulations of the PSR, and several sub-formulations of each of the two major formulations (my six below are by no means exhaustive). The first major formulation is in terms of explanation and the second is in terms of causation. Edward Feser, a contemporary Thomist, discusses both in Scholastic Metaphysics. In his terminology, he calls the explanatory version the PSR and the causal version the principle of causality (PC). I'll do the same, and I think it may be the PC that Thomists in particular are interested in.
The PSR comes in several sub-formulations, in terms of true propositions, beings, and events (and not everyone who accepts one formulation will accept the others):
- For any true proposition, p, there is a sufficient explanation for the truth of p
- For any existing being, b, there is a sufficient explanation for why b exists
- For any event, e, there is a sufficient explanation for why e occurs
The PC comes in several sub-formulations too:
Whatever is a compound of act and potency (i.e. has actual properties, but also has properties that it could potentially have, but does not actually have), has an external, already actual cause for why some properties become actualized, and some merely remain as potential properties. (See Edward Feser's Five Proofs for the Existence of God, chapter titled The Aristotelian Proof, and his book Scholastic Metaphysics, the section titled 2.3 The Principle of Causality for more on this.)
Whatever has its existence distinct from its essence, must have been caused to exist by something external to it. (See Edward Feser's Five Proofs for the Existence of God, the chapter titled The Thomistic Proof.)
Whatever begins to exist, must have an external cause of its coming into existence. (This is from William Lane Craig's Kalam cosmological argument.)
There are two noteworthy differences between the two version:
In the PC not everything has a cause; only things that are compounds of act and potency, essence and existence, or that begin to exist have a cause. In the explanatory PSR, yes everything (true propositions, beings, events) does indeed have an explanation.
In the PC, causes are always external to the thing being caused, since the idea of something being self-causing is absurd. But in the explanatory PSR, things can indeed be self-explaining (such as necessary beings, or beings for which there exists a successful ontological argument).
These two points are important to keep in mind because in your question you phrase the PSR as "Everything that exists must exist by something". If we're talking about the causal PC, then as Geremia says, no Thomist would accept this because God is something which is uncaused. If we're talking about the explanatory PSR, then "Everything that exists must exist by something" is true, but that is because we're taking about explanation and some things can be self explaining, namely God (but he's certainly not self-causing).
Getting back to your actual question...
In Scholastic Metaphyics, Feser considers four arguments for the (act-potency version of the) PC: appeals to self-evidence, empirical arguments, arguments from the PNC (principle of non-contradiction), and arguments from the PSR (the explanatory version). The arguments from self-evidence and empirical arguments are basically as you would expect. As for the argument from the PNC, he agrees with you that the PNC does not imply the PC, and writes "Some Thomists explicitly reject attempts to argue from PNC to PC." He notes that there doesn't seem to be an explicit contradiction in negating the PC, and attempts to derive a contradiction can be defended against, by the very act-potency distinction that Thomists want to embrace. Feser writes:
Now if we think of causation as essentially a matter of there first being a moment when a thing in no way has being, and then a later moment when it has being, then it is indeed hard to see any outright contradiction in the idea that this transition might lack a cause. But for the Scholastic that is the wrong way to characterize the situation. We should think instead of a thing’s potency for existence (which is not nothing even if it is not actual) being actualized at any particular instant it exists (and not merely by a temporally precedent cause).
Now seen in this light it may not seem so clear that denying PC does not involve a contradiction. For if the critic of PC is saying that a thing’s potential for existence can be actualized at a given instant without there being anything that does the actualizing, does that not entail that he is saying that the thing is at that instant both potential and actual with respect to its existence? And is that not a contradiction?
But that inference too would be too quick. For of course, the Scholastic himself says that a thing is at any instant both potential and actual. There is no contradiction here, because a thing is potential and actual in different respects. It is in potency with respect to its essence, but in act with respect to its existence. Now the critic of PC, it seems, can appeal to this very difference in order to defend himself against the charge of contradiction.
Finally, he offers an argument for the PC from the explanatory PSR. (Note, that Feser himself rejects the true-proposition version of the PSR, and where explanation is understood as implication so that the explanans logically imply the explanandum. He understands explanation as "making something intelligible") He writes:
So the argument from PNC to PC appears to fail. However, a more
popular approach among Scholastic writers to demonstrating PC is to
appeal to the principle of sufficient reason. (Cf. Gardeil 1967, pp.
22728; Phillips 1950, pp. 235-37; Renard 1946, p. 125-27) PSR states
that “everything is intelligible” (Garrigou-Lagrange 1939, p. 181),
and that “there is a sufficient reason or adequate necessary objective
explanation for the being of whatever is and for all attributes of any
being” (Wuellner 1956b, p. 15). But if PC were false -- if the
actualization of a potency, the existence of a contingent thing, or
something’s changing or coming into being could lack a cause — then
these phenomena would not be intelligible, would lack a sufficient
reason or adequate explanation. Hence if PSR is true, PC must be true.
PC is an application of PSR to things that are mixtures of act and
potency and essence and existence, and which therefore – unlike God,
who as pure actuality and subsistent being itself has the sufficient
reason or adequate explanation for his existence within himself –
require an explanation by reference to something outside them.