Depends on the requirements of the web and desktop app.
MySQL is a transactional database so it will be able to handle multiple requests from both web and thick client applications.
To be honest you would be better looking into some of the fantastic Open Source CRM offerings like OpenCRX, vTiger, or Tine 2.0
vTiger is probably the pick of the 3 above but it already has capabilities you want, you won't need to learn MySQL as much and get on with configuration and use of the system.