Learn Russian: Ввиду огромного ущерба мы поддерживаем этот запрет. - In view of the huge damage, we support this ban.

When it functions as a formal preposition meaning 'in view of' or 'due to', it is written as one word: ввиду. You write it as two words (в виду) only when using the literal noun вид (view/sight), most famously in the set phrase иметь в виду (to keep in mind / to mean).

What case is огромного ущерба, and what triggers it?

It is in the Genitive case, triggered by the preposition ввиду. Formal compound prepositions of cause, such as ввиду and вследствие, always require the noun that follows them to take the Genitive case.

Could we use вследствие instead of ввиду in this sentence?

Yes, mechanically both take the Genitive case, but there is a slight difference in nuance. Ввиду ('in view of') is often used for existing circumstances that motivate a decision, like supporting a ban. Вследствие ('as a consequence of') emphasizes a direct chain of events stemming from a past cause. Both work well here in formal contexts.

Why does the direct object этот запрет look exactly like its dictionary form?

It is in the Accusative case because it receives the action of the verb поддерживаем (we support). However, because запрет (ban) is an inanimate masculine noun, its Accusative form is identical to its Nominative form, so the ending does not change.