Sic discipuli discunt ordinem, vigilias, et animos firmos et in bello et in pace magnam vim habere.

Because after discunt Latin is using an indirect statement construction.

So instead of saying:

• discipuli discunt quod ... habent = the students learn that ... have

Latin very often says:

• discipuli discunt ... habere = literally, the students learn ... to have, but idiomatically the students learn that ... have

So habere is not the main verb of the sentence. The main verb is discunt.
The infinitive habere introduces what the students are learning.

They are accusative because in an indirect statement, the subject of the infinitive is put in the accusative case.

So in this sentence:

• ordinem
• vigilias
• animos firmos

are the things that are understood as the subject of habere.

In clearer English structure, the sentence means something like:

• The students learn that order, watches, and steadfast spirits have great power...

In Latin, those “subjects” of have appear in the accusative because they belong to the infinitive construction.

Yes. It is a list of three things that all go together with habere:

• ordinem = order / discipline
• vigilias = watches / sentry-duty / wakefulness
• animos firmos = steadfast spirits / brave hearts / firm courage

All three are being said to have great force.

So the structure is:

• discunt
  ordinem, vigilias, et animos firmos
  magnam vim habere

= they learn that order, vigilance, and steadfast spirits have great power

Because animos firmos is part of the accusative subject of the indirect statement.

Since animos is accusative plural masculine, the adjective firmos has to agree with it:

• animos = accusative plural masculine
• firmos = accusative plural masculine

If Latin had used firmi animi, that would be nominative plural, which would not fit this construction.

So animos firmos means firm spirits or steadfast hearts, and both words are accusative because of the indirect statement.

Magnam vim is the direct object of habere.

• vim = force, power, strength, influence
• magnam = great

So:

• magnam vim habere = to have great power / to have great force

The three accusative nouns in the list are the subject of habere, and magnam vim is what they “have.”

So the internal structure is:

• ordinem, vigilias, et animos firmos = subject of the infinitive
• magnam vim = object of the infinitive
• habere = infinitive verb

This is a very common Latin pattern: et ... et ...

It means both ... and ...

So:

• et in bello et in pace = both in war and in peace

Latin often repeats the conjunction this way for balance and emphasis.

You could think of it as stronger and more formal than a single et.

Because in with the ablative usually means in, on, or during a place or condition, without motion into it.

Here there is no movement. The phrase means:

• in bello = in war
• in pace = in peace

So Latin uses in + ablative.

By contrast, in + accusative usually means movement into something:

• in urbem = into the city

That is not the idea here.

Sic means thus, so, or in this way.

It usually connects the sentence to what came before. It signals that the students learn this lesson in this way or thus as a result of what has just been described.

So sic is not changing the grammar of the sentence; it is helping the flow of the argument or narrative.

It is the present tense of disco, so depending on context it can mean:

• learn
• are learning
• sometimes even come to learn

Latin present tense is often broader than English present tense.
So the exact translation depends on context.

In this sentence, discunt most naturally means:

• the students learn or
• the students are learning

Both can be correct depending on the passage.

Vigilias is the accusative plural of vigilia.

In military or disciplined contexts, vigiliae often means:

• watches
• guard duty
• keeping watch
• periods of wakefulness / vigilance

So here it probably refers not just to “staying awake” in a general sense, but to disciplined alertness, especially the sort valued in military life.

That fits well with:

• ordinem = order / discipline
• vigilias = watchfulness / guard-duty
• animos firmos = steadfast courage

Latin word order is much freer than English word order because the endings show how words function.

English depends heavily on position:

• The students learn that order has great power.

Latin can move things around for emphasis or style:

• Sic discipuli discunt ordinem ... magnam vim habere.

A few things to notice here:

• discipuli comes early as the subject of discunt
• the list of important ideas comes next: ordinem, vigilias, et animos firmos
• magnam vim comes just before habere
• habere is saved for the end, which is very natural in Latin

So the word order is not random; it is stylistic and emphatic rather than rigidly fixed as in English.

A good way is to divide it into layers.

First find the main clause:

• Sic discipuli discunt = Thus the students learn

Then find what they learn: the indirect statement:

• ordinem, vigilias, et animos firmos ... magnam vim habere

Then break that indirect statement apart:

• ordinem, vigilias, et animos firmos = accusative subject
• magnam vim = object
• habere = infinitive verb
• et in bello et in pace = adverbial phrase modifying habere or the whole idea of having great power

So the whole thing is:

• Thus the students learn [that order, watches, and steadfast spirits have great power both in war and in peace].

That is the clearest way to parse it.