Learn French: Que le vétérinaire vaccine le chien est essentiel. - That the vet vaccinates the dog is essential.

Here, Que translates to "That" and introduces a whole clause (le vétérinaire vaccine le chien) acting as the subject of the sentence. You can think of it as an inverted, more formal version of saying "It is essential that...".

Is "vaccine" in the subjunctive here? It looks exactly like the normal present tense.

Yes, it is in the subjunctive! The main idea of the sentence is that something est essentiel (is essential). Expressions of necessity always trigger the subjunctive in the dependent clause. For regular -er verbs like vacciner, the subjunctive il/elle form happens to be spelled exactly the same as the present indicative.

If I want to say this in everyday conversation, is starting with "Que" the most natural way?

Not really. Starting a sentence with a Que clause is grammatically correct but sounds quite formal or literary. In everyday spoken French, you would normally use an impersonal structure instead, such as Il est essentiel que le vétérinaire vaccine le chien or C'est essentiel que....

Why doesn't the adjective "essentiel" agree with any of the nouns in the sentence?

The adjective essentiel isn't describing le vétérinaire or le chien; it is describing the entire concept of the dog getting vaccinated. When an adjective modifies a whole clause or an idea rather than a specific noun, it always takes the default masculine singular form.