Learn French: Le responsable n'autorise que les clients VIP.

The structure ne ... que is a restriction that means 'only' or 'nothing but'. It wraps around the verb similar to a negative structure, but it doesn't negate the action. Here, n'autorise que means 'only allows'.

Can I use seulement instead of ne ... que here?

Yes, you could completely drop the n' and say Le responsable autorise seulement les clients VIP. Both mean exactly the same thing, but the ne ... que structure is extremely common in everyday French.

Doesn't responsable mean 'responsible'? Why is it translated as 'manager'?

While it is the direct translation of the adjective 'responsible', it is very frequently used as a noun (le responsable or la responsable) to mean the person in charge. Depending on the context, this translates to manager, supervisor, or director.

Does the acronym VIP need an s to match the plural clients?

No, acronyms used as adjectives, especially those borrowed from English like VIP, are generally invariable in French. It is also usually pronounced with the English letter sounds instead of the French ones.