There is no escape. This chapter is a primer on the claim. Next week a difficult chapter on the science of it all. But do not miss the comment contra locality in physics in the penultimate paragraph.
1 As a consequence, it is clear that God must be everywhere and in all things.
2 For, the mover and the thing moved must be simultaneous, as the Philosopher proves. But God moves all things to their operations, as we have shown. Therefore, He is in all things.
3 Again, everything that is in a place, or in something, is in some way in contact with it. For instance, a bodily thing is in place in something according to the contact of dimensive quantity; while an incorporeal thing is said to be in something according to the contact of power, since it lacks dimensive quantity. And so, an incorporeal thing is related to its presence in something by its power, in the same way that a corporeal thing is related to its presence in something by dimensive quantity. Now, if there were any body possessed of infinite dimensive quantity, it would have to be everywhere. So, if there be an incorporeal being possessed of infinite power, it must be everywhere. But we showed in Book One  that God is of infinite power. Therefore, He is everywhere.
Notes Against the idea any material thing is actually infinite: “if there were any body possessed of infinite dimensive quantity, it would have to be everywhere.”
4 Besides, as a Particular cause is to a particular effect, so is a universal cause to a universal effect. Now, a particular cause must be simultaneous with its proper particular effect. Thus, fire heats through its essence, and the soul confers life on the body through its essence. Therefore, since God is the universal cause of the whole of being, as we showed in Book Two , it must be that wherever being is found, the divine presence is also there.
5 Moreover, whenever an agent is present only to one of its effects, its action cannot be transferred to another, unless by using the first effect as an intermediary, because the agent and the patient must be simultaneous. For instance, the organic motive power does not have a member of the body except through the heart as an intermediary. So, if God were present to but one of His effects—for instance, to the first moved sphere which would be moved immediately by Him—it would follow that His action could not be transferred to another thing except through the mediation of this sphere.
Now, this is not appropriate. Indeed, if the action of any agent cannot be transferred to other things except through the mediation of a first effect, then this effect must correspond proportionally with the agent according to its entire power; otherwise, the agent could not use his entire power. We see an instance of this in the fact that all the motions that the motive power can cause can be carried out through the heart.
But there is no creature that can serve as a medium for the carrying out of whatever the divine power can do, for divine power infinitely surpasses every created thing, as is evident from the things shown in Book One . Therefore, it is not appropriate to say that divine action does not extend to other effects except through the mediation of a first one. So, He is not merely present in one of His effects, but in all of them. The same reasoning will be used if a person says that He is present in some and not in others, because, no matter how many divine effects are taken, they could not be sufficient to carry out the execution of the divine power.
6 Furthermore, an agent cause must be simultaneous with its proximate and immediate effect. But there is in everything a proximate and immediate effect of God Himself. For we showed in Book Two  that God alone can create. Now, there is in everything something caused by creation: prime matter in the case of corporeal things, in incorporeal things their simple essences, as is evident from the things that we determined in Book Two [15ff]. Therefore, God must be simultaneously present in all things, particularly since He continually and always preserves in being those things which He has brought into being from nonbeing, as has been shown.
7 Hence it is said: “I fill heaven and earth” (Jer. 23:24); and in the Psalm (138:8): “If I ascend into heaven, You art there; if I descend into hell, You art present.”
Notes Wait, is this pantheism? No, friend. Read on.
8 Through this conclusion, moreover, the error is set aside of those who say that God is in some definite part of the world (for instance, in the first heaven and in the eastern section) and that He is consequently the principle of heavenly motion.
Of course, this statement of theirs could be supported, if soundly interpreted: not, for instance, that we may understand God as being confined to some determinate part of the world, but that the source of all corporeal motions, according to the order of nature, takes its start from a determinate part, being moved by God.
Because of this He is spoken of in Sacred Scripture also as being in the heavens in a particular way; in the text of Isaiah toward the end (66:1): “Heaven is My throne,” and in the Psalm (113: 16): “The heaven of heaven is the Lord’s,” and so on.
But from the fact that, apart from the order of nature, God performs some operation in even the lowest of bodies which cannot be, caused by the power of a celestial body it is clearly shown that God is immediately present, not only in the celestial body, but also in the lowest things.
9 But we must not think that God is everywhere in such a way that He is divided in various areas of place, as if one part of Him were here and another part there. Rather, His entire being is everywhere. For God, as a completely simple being, has no parts.
10 Nor is His simplicity something like that of a point, which is the terminus of a continuous line and thus has a definite position on this line, with the consequence that one point is impossible unless it is at one, indivisible place. In fact, God is indivisible, in the sense of existing entirely outside the genus of continuous things. And so, He is not determined in regard to place, either large or small, by any necessity of His essence requiring Him to be in a certain place, for He has been from eternity prior to all place. But by the immensity of His power He touches upon all things that are in place, for He is the universal cause of being, as we said. Thus, He is present in His entirety wherever He is, since He touches upon all things by His simple power.
Notes This, I believe, is the locus of arguments against locality. Think EPR, Bell, and all that. Locality cannot be maintained. Nor need it be, as long as we recognize there is always a First Cause.
11 Yet, we must not think that He is present in things, in the sense of being combined with them as one of their parts. For it was shown in Book One [17, 27] that He is neither the matter nor the form of anything. Instead, He is in all things in the fashion of an agent cause.