Can I offer my customers hotels where they pay at the hotel instead of paying me up front?

Often yes, but only on the rates that allow it. A hotel booking API usually returns a mix. Some rooms are prepaid, which means you take the money from your customer at the time of booking and settle with the supplier afterwards. Others are pay at hotel, where your customer pays nothing to you and settles the room bill at the reception desk when they check out.

A pay at hotel room still needs a card at the time of booking, because the hotel wants a guarantee before it holds the room. That card is passed through to the hotel to hold rather than charged by you, and the hotel may place a temporary block on it. Whether you get this option at all depends on the supplier you are connected to. Some suppliers sell only prepaid rooms, so you will never see a pay at hotel rate from them, whatever the hotel offers on its own website.

It is worth thinking about what this does to your money. When your customer pays the hotel directly, nothing passes through your account, so your commission reaches you later from the supplier instead of being in your hands on the day of the booking. Some agencies prefer prepaid rates for that reason alone, and you can usually choose to show only prepaid rooms if you would rather keep it simple.

Ask your supplier which of their rates can be paid at the hotel, then check how the payment type appears on your search results so your customer knows exactly what they are paying now and what they are paying later.