Learn Turkish: Herhangi bir şube uygun.

Yes, when you are modifying a noun to mean 'any [noun]', you must use the full phrase herhangi bir. If you just say herhangi şube, it sounds grammatically incomplete to a Turkish speaker.

In English we sometimes say 'any branches'. Can I make şube plural here and say herhangi bir şubeler?

No, a noun following herhangi bir must always be in the singular form in Turkish. This is a strict rule that also applies when you use singular nouns after numbers or words like birkaç (a few).

Where is the verb 'is' in this sentence?

In Turkish, sentences that describe a noun with an adjective often don't need an explicit verb for the third person ('is'). The adjective uygun (suitable) acts as the whole predicate here. You could add the 'to be' suffix -dur to make it uygundur, but it's extremely common to drop it in everyday speech.

Does şube mean a branch of a tree, or just a business branch?

Şube is only used for organizational or business branches, like a bank branch, a post office, or a local store in a larger chain. If you want to talk about the physical branch of a tree, you would use the word dal instead.