Can I see which hotels are currently sold out through the API so I know not to show them to my customers?

No, the hotel booking API will not tell you ahead of time which specific properties are sold out on a given date. What you can do instead is search for hotels in a destination for your customer's travel dates, and the API will return only the hotels that have rooms available at that time. Any hotel that does not appear in the search results simply has no inventory left to sell, so you will naturally avoid offering something you cannot book.

The practical way to handle this is to search through the API whenever a customer asks about hotels for a particular date range. This gives you a real-time picture of what is actually available on that day. If a hotel was available yesterday but is not there today, it means the rooms have sold out. Some hotels can fill up quickly during busy periods, so searching at the moment your customer is ready to book gives you the most accurate information.

Keep in mind that availability changes constantly because other travel agencies and customers are booking at the same time. A hotel that shows as available when you search might only have one or two rooms left. If your customer takes a long time to decide, that last room could sell to someone else before they commit to the booking. This is why many travel agencies encourage customers to book quickly when they find something they like, rather than sitting on a decision for days.

To give your customers the best experience, search for hotels in their destination and dates as soon as they tell you their travel plans, and show them what actually came back from the API rather than trying to predict what will or will not be available.