Les deux formulations sont correctes.

1. Vous irez visiter ces îles exotiques à condition d’avoir votre visa.
2. Vous irez visiter ces îles exotiques à condition que vous ayez votre visa.

In your example, both are correct because the subject of the main sentence is the same as the subject of the subordinate condition (which is required to be able to use "de"). When the subjects are different, you have to use "que":

2. Vous irez visiter ces îles exotiques à condition que l’ambassade vous délivre un visa.

While the second form is grammatically correct and à condition que + le/la/les is common, à condition d’avoir "sounds better" and is in my opinion what most native speaker would chose.

Your question only appears in a 2012 Polish academic MCQ where you have to chose the "right" answer therefore I have absolutely no doubt about the fact à condition d’avoir un visa is the expected answer.

This is confirmed by this page which list it as correct.

To summarize, if you are a student learning French and using these tests as training material, I would recommend to memorize that à la condition d’avoir is the expected answer.

On the other hand, if you are a student who chose the à condition que vous ayez answer but a teacher counted you wrong, you can definitely challenge the decision since that form is not incorrect.

Note also that the indicative is also technically possible (see for example the TLFi) leading to:

Vous irez visiter ces îles exotiques à condition que vous avez votre visa.

Oui Monsieur, j’y consens bien volontiers, à la condition que vous viendrez dîner chez moi demain. Maupassant, Notre cœur, 1890.

but nowadays, it would be hard to find someone not startled by it. The OQLF even rejects the indicative.

“… à condition d’avoir …” is correct. The second sentence is grammatically correct, but not idiomatic at all.

When a verb can be constructed with either “de … [noun phrase]” or “que … [verb phrase]”, if the subject of the verb phrase is the same as the subject of the main clause, you must replace the verb phrase by de followed by an infinitive.

Not using the infinitive in this sentence feels to me (native speaker from France) as extremely outdated. It’s something you might read in a 19th century (maybe even 18th century) book, but not something I would say. It may have survived in some local usage, but I don’t remember encountering this often. If I heard it, I’d assume that the speaker lost his/her train of thought and forgot exactly how the sentence started.

I’m very surprised to see a native speaker suggest otherwise.