Bella, we stayed at the Links last year, and could not pick up any wifi near there. There are a few restaurants, cafes, that offer free wifi. A couple that I remember are "Salt and Pepper" near Playa Linda, and Charlies Bar in or near San Nicolas. Normally we would go for coffee in the morning to Salt and Pepper, and I would check emails once a day. You can check with the concierge, I'm sure there are a lot more establishments around the island with free wifi.
We will be at the Links next week, I'll give you an updated list when we return.
Tom R