If an employee books a flight that costs more than their company allows, can their manager be asked to approve it before the ticket is issued?

Yes, that is possible. A corporate booking portal can hold the booking instead of ticketing it straight away and send it to the person the company has named as the approver. You set the rules for each company you handle, such as a ceiling on the fare, a cabin class the staff are allowed to fly, or a rule that anything booked within a few days of travel needs a second pair of eyes. When a booking stays inside those rules, it goes through on its own. When it breaks one, it stops and waits.

The employee sees the same search and the same fares, so nothing changes for them until they confirm. At that point they get a message saying the booking is waiting for approval, and the approver gets an email with the trip details, the fare, and the reason it was flagged. The approver can allow it or refuse it, and the reply is recorded against the booking so the company can look back later and see who cleared what.

Be honest with your corporate clients about one thing. A held fare is not a frozen fare. Airlines and consolidators only guarantee a price for a short window, and some fares cannot be held at all, so if the approver takes a day to reply, the seat or the price may have moved by then. Whether the system can genuinely park a live fare, and for how long, depends on the supplier behind that particular booking, not on the portal.

Before you offer this to a company, ask them who their approvers are, what should happen when an approver is on leave, and how long they are willing to let a booking sit before it is released.