How do the Seals relate to the Trumpets and the Vials?
The Seven Seals are not events in themselves but a kind of table of contents that previews what is written inside the sealed book; once the seventh seal is opened, the book itself unfolds as the actual Trumpet Judgments and Vial Judgments that occur on the earth.
Revelation 5 shows the Lamb receiving a book that is sealed shut with seven seals, written both within and on the back. As each seal is broken, it does not release an event onto the earth, it simply reveals a foreshadowing of what is contained inside the book, much like a Table of Contents with descriptions. The seals introduce the major themes, agents, and outcomes that the book itself will later unfold in detail.
Once all seven seals are opened, the seventh seal brings silence in heaven and the seven angels with trumpets prepare to sound. At that point the book is fully open, and from Revelation 8 onward you are no longer looking at previews but at the actual experiences that take place on the earth. So the seals summarize, and the Trumpets and Vials are the real, unfolding fulfillment of what those seals pointed to.
The Trumpet Judgments come first: hail, fire, and blood burning up the grass and a third of the trees, a burning mountain turning a third of the sea to blood and destroying sea life and ships, and further trumpets that follow in sequence. After the trumpets run their course, the Vials of Wrath are poured out, carrying forward the judgments referenced under the seals to their conclusion. The sixth seal in particular ties directly to this sequence, showing the great earthquake, the sun darkening, the moon turning to blood, and the stars falling, and these same elements are matched to specific trumpets and vials, for example the great earthquake appears again at the sixth and seventh trumpets and the seventh vial, and the darkened sun and bloodied moon appear again at the fourth and fifth trumpets and the fifth vial.
So the relationship is this: the seals are the announcement, the trumpets and vials are the fulfillment. The seals foreshadow judgment, wrath, martyrdom, and the eventual establishing of Christ's kingdom, and it is the trumpets and vials that actually carry those forecasted judgments out on the earth, leading at last to the reign of Christ.
The scriptures
And I saw in the right hand of him that sat on the throne a book written within and on the backside, sealed with seven seals.
And when I looked, behold, an hand was sent unto me; and, lo, a roll of a book was therein; And he spread it before me; and it was written within and without: and there was written therein lamentations, and mourning, and woe.
And when he had opened the seventh seal, there was silence in heaven about the space of half an hour. And I saw the seven angels which stood before God; and to them were given seven trumpets. And another angel came and stood at the altar, having a golden censer; and there was given unto him much incense, that he should offer it with the prayers of all saints upon the golden altar which was before the throne. And the smoke of the incense, which came with the prayers of the saints, ascended up before God out of the angel’s hand. And the angel took the censer, and filled it with fire of the altar, and cast it into the earth: and there were voices, and thunderings, and lightnings, and an earthquake. And the seven angels which had the seven trumpets prepared themselves to sound.
The first angel sounded, and there followed hail and fire mingled with blood, and they were cast upon the earth: and the third part of trees was burnt up, and all green grass was burnt up.
And the second angel sounded, and as it were a great mountain burning with fire was cast into the sea: and the third part of the sea became blood; And the third part of the creatures which were in the sea, and had life, died; and the third part of the ships were destroyed.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.