Learn Hardcore Ukrainian: Інтернет-магазин обговорює обмін товару. - The online store discusses a product exchange.

Because it is a compound noun: інтернет + магазин. In Ukrainian, combinations like інтернет-магазин, пресслужба, бізнес-план are often written as single lexical units, and інтернет-магазин is standard with a hyphen.

In your sentence it is capitalized only because it stands at the beginning of the sentence. In the middle of a sentence, you would normally write інтернет-магазин.

What case is Інтернет-магазин in?

It is in the nominative singular. It is the subject of the sentence—the thing doing the action.

інтернет-магазин = nominative singular
masculine noun
inanimate

So grammatically it works like the online store as the subject.

Why is обговорює ending in -є?

Обговорює is the 3rd person singular present tense form of обговорювати (to discuss).

So:

я обговорюю = I discuss / am discussing
ти обговорюєш = you discuss
він / вона / воно обговорює = he / she / it discusses

Since the subject інтернет-магазин is singular, the verb is also singular: обговорює.