No, you needn’t show any proof. The only time it’s an issue is if you travel from Ecuador or Colombia to Costa Rica. That government requires proof of yellow fever vaccination. THE CDC has a spectrum of vaccinations it suggests for travel anywhere in the tropics — your choice whether you get them or not. We did Hep A/B long ago. Malaria is no longer an issue in Ecuador, by the way. Dengue can be a problem in coastal issues (as it is everywhere in the tropics), so as you say, if you go to the coast, use repellant, screens on the windows, etc. And maybe it happens, but I’ve never heard of anyone coming down with yellow fever…