Can a trip wait for the manager's approval before it actually gets booked for my corporate client?

Yes, an approval step can sit in the middle of the booking process so nothing is confirmed until the right person says yes. The employee searches and picks the flight or hotel as normal, but instead of going straight to payment the request is held and sent to their approver. Only once it is approved does the booking go through to the supplier.

How strict you make it is up to the company. Approval can be required for every trip, or only when certain conditions are met, such as a trip over a spend limit, a long haul destination, a higher class of travel, or an option that sits outside the company's travel policy. Anything inside policy can be allowed to book straight away so your corporate client's staff are not waiting around for routine trips. The approver usually gets an email or a message with a link, can see what was chosen and why, and can approve or decline with a note. More than one level of approval can be set up where a company needs it, and reminders can be sent if the approver does not reply.

The honest limit here is the fare itself. While a request is sitting with an approver, the airline or hotel is not holding that price for you, and on many fare types there is no way to hold it at all. Whether a short hold is possible depends entirely on the supplier and the fare, so a slow approval can mean the price or the seat has moved by the time it is released. The system can re-check and show the new price rather than book something the employee did not agree to.

Before setting this up, ask your corporate client who should approve what, how many levels they really need, and which trips can skip approval altogether, because a workflow that is too tight will simply push their staff into booking elsewhere.