Booking/reservation site

Hello,

I have a problem with trying to create booking/reservation site for my customer,

My customer has a company which provides taxi drivers for different taxi companies. Those companies can also add available cars to the system.

What I would need is

1. Simple and effective private site (with login) with functionality (for drivers: book a shift / for companies: add a shift). Site would need to have two type of registration-levels, one for drivers and the other one for companies who add available cars to the system.

2. Calendar with detailed daily view of available shifts for drivers. View should be as listing with options to arrange by different categories, for example by time, car, type of car, company etc. View should open directly from clicking the date on calendar.

3. Ability for companies to add job-opportunities

When the driver books a shift from the system the shift should be unbookable by any other employee.

I have seen one system like that made with Joomla but I’m not so familiar with Joomla so my question is it possible to make site like that with WordPress?