Can the flight booking API search a few days either side of my customer's date so I can show them a cheaper day to fly?

Yes, in many cases, although how wide a spread you can look at depends on the supplier sitting behind the search. A flexible date search asks for the same route across a range of departure days instead of one fixed day, and hands back the lowest fare found on each of those days, so you can put a small calendar in front of your customer instead of a single result. Some suppliers offer this properly as part of their service, while others will only answer one date at a time.

Where the supplier does not support it, the same effect can still be built up by running several ordinary searches behind the scenes and putting the answers together. That works, but it is slower and much heavier on the connection, so the window is normally kept to a few days on either side rather than a whole month. If your customer wants to compare a long stretch of dates, expect the page to take noticeably longer to fill.

The thing to be careful about is that calendar prices are often drawn from cached or indicative fares rather than a live quote taken at that moment. They are there to point your customer towards the cheaper days, not to act as a promise. Once a date is chosen, the flight should be priced again properly before you take any payment, and the fare can move between those two steps.

Check whether the suppliers connected to your account offer a flexible date search today, and if they do, agree with your team how wide a window you want to show so that searches stay quick for everyone using your site.