Building up an IVR would be possible.
But: What's the actual infrastructure we're starting with ?
And for the point: "Make a booking" How would You like this to be processed ? Usually You will need informations like a pickup-point or/and a pickup-time. This vcould be processed by audio-recording and fowarding the audio-results to an operator but this is not an integrated solution. An integrated solution could probably be achieved by using Voice-Recognition but this is in no way possible within Your budget.
All tasks in regard to booking status (incl. connect to the driver, serving the booking) would require access to Your dispo-system to check the status and realtions for a booking of the caller (which is only identified by either a telephone-number or a booking-code which will be presented to the caller once he placed a booking.
You see, there are still a lot of open questions which will leaad us to an effort excessing Your planned budget.
Nevertheless we may start with the project and have a look, how far it could be driven :)