Can I hold a hotel room for my customer for a day or two while they make up their mind, without paying for it?

Sometimes, but not on every hotel, and it depends on the supplier the rate comes from. Most hotel rates that reach your portal through an API are live rates, which means the room is only reserved at the moment the booking is confirmed. Until that happens the room stays in general sale, so another agent or another website can take it while your customer is still deciding.

The way most travel businesses handle this is with free cancellation rates rather than a hold. You confirm the booking for your customer straight away, and if they change their mind before the cancellation deadline you cancel it at no cost. Your customer gets a genuinely confirmed room and you are not locked in. The catch is that the deadline is set by the hotel or the supplier, not by the software, and some rates are non refundable from the moment they are booked, so this will not work on every option in your search results.

A smaller number of suppliers do offer a proper option period, where a room is held for a day or two before any payment is made. That is usually on contracted or group rates rather than ordinary online rates, and it is often arranged by the supplier's own team instead of through the API. Whether you can use it comes down to which suppliers are connected to your portal and what your agreement with them allows.

If holding rooms matters to the way you sell, check the cancellation terms shown against each rate in your search results, and ask which of your connected suppliers will give you an option period on group bookings.