Learn Italian: In estate potete riposare tranquillamente.

To form an adverb with -mente, you take the feminine singular form of the adjective and add the suffix. The adjective for "peaceful" or "quiet" is tranquillo. First, change it to its feminine form (tranquilla), and then add -mente to get tranquillamente.

Why is it in estate instead of nell'estate?

When talking generally about something that happens during a season, Italian simply uses the preposition in without a definite article. So, "in summer" translates directly to in estate.

Why is riposare left in its dictionary form instead of being conjugated?

Because it follows potete (you all can), which is a modal verb. When a modal verb is paired with another action, only the first verb gets conjugated. The second verb always stays in its infinitive (dictionary) form.

I have seen "to rest" written as the reflexive verb riposarsi. Why isn't it reflexive here?

Both riposare and the reflexive riposarsi mean "to rest" and they are often interchangeable. You could also say potete riposarvi tranquillamente here, and the meaning would be exactly the same!