According to Trinitarians, can a Person of the Trinity add a new nature whenever He wants to?

Yes, a Divine Person can assume multiple human natures (Summa Theologica q. 3 a. 7 co.):

What has power for one thing, and no more, has a power limited to one. Now the power of a Divine Person is infinite, nor can it be limited by any created thing. Hence it may not be said that a Divine Person so assumed one human nature as to be unable to assume another. For it would seem to follow from this that the Personality of the Divine Nature was so comprehended by one human nature as to be unable to assume another to its Personality; and this is impossible, for the Uncreated cannot be comprehended by any creature. Hence it is plain that, whether we consider the Divine Person in regard to His power, which is the principle of the union, or in regard to His Personality, which is the term of the union, it has to be said that the Divine Person, over and beyond the human nature which He has assumed, can assume another distinct human nature.

^{cited on p. 252 of The Thomist 65 (2001): 239-58, "Aquinas on Intelligent Extra-Terrestrial Life"}

Human nature is more assumable (Summa Theologica III q. 4 a. 1 co.)

1. According to its dignity
   because human nature, as being rational and intellectual, was made for attaining to the Word to some extent by its operation, viz. by knowing and loving Him.

2. According to its need
   because it stood in need of restoration, having fallen under original sin.

Now these two things belong to human nature alone. For in the irrational creature the fitness of dignity is wanting, and in the angelic nature the aforesaid fitness of need is wanting. Hence it follows that only human nature was assumable.

According to Trinitarians, can a Person of the Trinity add a new nature whenever He wants to?

The simple answer is no!

When God spoke to Moses he said ”I Am that I Am” (אֶהְיֶה אֲשֶׁר אֶהְיֶה‎‎). He never said: ”I am who I am; however hold on, I can change who I am when I want.” To do so would demonstrate that God is not united within the Trinity. Divine Simplicity would cease to exist!

Divine simplicity is central to the classical Western concept of God. Simplicity denies any physical or metaphysical composition in the divine being. This means God is the divine nature itself and has no accidents (properties that are not necessary) accruing to his nature. There are no real divisions or distinctions in this nature. Thus, the entirety of God is whatever is attributed to him. Divine simplicity is the hallmark of God’s utter transcendence of all else, ensuring the divine nature to be beyond the reach of ordinary categories and distinctions, or at least their ordinary application. Simplicity in this way confers a unique ontological status that many philosophers find highly peculiar.

Not answering this question to create an endless discussion on this subject matter. But according to St. Thomas Aquinas, God’s nature means that God, from whom everything else is created, “contains within Himself the whole perfection of being” (ST Ia 4.2). But as the ultimate cause of our own existence, God is said to have all the perfections of his creatures (ST Ia 13.2). Thus, if God contains the ultimate in perfection and is complete within the Sacred Trinity, it makes no sense that God could add other Person to the Trinity or add a new nature whenever He wants to? Stability is what the Divine Trinity is. To say otherwise would be that God would be unstable and changeable.

God’s Nature

Once Aquinas completes his discussion of the theistic demonstrations, he proceeds to investigate God’s nature. Such an investigation poses unique challenges. Although Aquinas thinks that we can demonstrate God’s existence, our demonstrative efforts cannot tell us everything about what God is like. As we noted before, God’s nature—that is, what God is in himself—surpasses what the human intellect is able to grasp (SCG I.14.2). Aquinas therefore does not presume to say explicitly or directly what God is. Instead, he investigates divine nature by determining what God is not. He does this by denying of God those properties that are conceptually at odds with what is already concluded by means of the five ways (ST Ia 3 prologue; Cf. SCG I.14.2 and 3).

Aquinas acknowledges a potential worry for his view. If the method by which we investigate God is one of strict remotion, then no divine predicate can describe what God is really like. As one objection states: “it seems that no name can be applied to God substantially. For Damascene says … ‘Everything said of God signifies not his substance, but rather exemplifies what he is not; or expresses some relation, or something following his nature or operation” (ST Ia 13.2 ad 1). In other words, the terms we attribute to God either function negatively (for example, to say God is immaterial is to say he is “not material”) or describe qualities that God causes his creatures to have. To illustrate this second alternative: consider what we mean when we say “God is good” or “God is wise.” According to the aforementioned objection, to say that God is good or wise is just to say that God is the cause of goodness and wisdom in creatures; the predicates in question here do not tell us anything about God’s nature (Ibid.).

For Aquinas, however, the terms we predicate of God can function positively, even if they cannot capture perfectly or make explicit the divine nature. Here’s how. As we have discussed, natural knowledge of God is mediated by our knowledge of the created order. The observable facts of that order reveal an efficient cause that is itself uncaused—a self-subsisting first mover that is uncreated and is not subject to any change. According to Aquinas, this means that God, from whom everything else is created, “contains within Himself the whole perfection of being” (ST Ia 4.2). But as the ultimate cause of our own existence, God is said to have all the perfections of his creatures (ST Ia 13.2). Whatever perfections reside in us must be deficient likenesses of what exists perfectly in God. Consequently, Aquinas thinks that terms such as good and wise can refer back to God. Of course, those terms are predicated of God imperfectly just as God’s creatures are imperfect likenesses of him. “So when we say, ‘God is good,’ the meaning is not, ‘God is the cause of goodness,’ or ‘God is not evil’; but the meaning is, ‘Whatever good we attribute to creatures, pre-exists in God,’ and in a more excellent and higher way” (Ibid.).

Moreover, denying certain properties of God can, in fact, give us a corresponding (albeit incomplete) understanding of what God is like. In other words, the process of articulating what God is not does not yield an account of the divine that is wholly negative. Here is a rough description of the way Aquinas’ reasoning proceeds: we reason from theistic arguments (particularly the first and second ways) that God is the first cause; that is, God is the first being in the order of efficient causality. If this is so, there can be no potency or unrealized potential in God. For if something has the potential or latent capacity to act, then its activity must be precipitated by some prior actuality. But in this line of reasoning, there is no actuality prior to God. It must follow, then, that God is pure actuality, and this in virtue of being the first cause (ST Ia 3.1). So although this process denies God those traits that are contrary to what we know about him, those denials invariably yield a fairly substantive account of the divine life.

Other truths necessarily follow from the idea that God is pure actuality. For example, we know that God cannot be a body. For a characteristic feature of bodies is that they are subject to being moved by something other than themselves. And because God is not a body, he cannot be a composite of material parts (ST Ia 3.7). Not only does Aquinas think that God is not a material composite, he also insists that God is not a metaphysical composite (Vallencia, 2005). In other words, God is not an amalgam of attributes, nor is he a being whose nature or essence can be distinguished from his existence. He is, rather, a simple being.