I would recommend Asterisk. You can set up "hunt groups" so that Asterisk will try to route the call to one of your volunteers. However, you would need multiple phone lines, otherwise your callers will be greeted with a "busy" tone.
In the UK, there is a company called AQL which will provide you with multiple distinct telephone numbers, all of which can be routed over the internet (via AIX) to your Asterisk box. This solves the "busy" problem, but it does involve some cost. Perhaps there is a similar telecoms company in Pakistan?
If not, you will need to rent multiple phone lines, all of which will have to be physically connected to the Asterisk box. You will need two lines for each concurrent call you plan to handle - one for the incoming call, and one for the outgoing call to the volunteer's mobile.
Keep in mind you will also need to pay for the calls to your volunteers' mobile phones - if they have suitable internet connections then a SIP softphone (or Skype) would remove the cost of outgoing mobile calls.
In fact, doing this via Skype could be quite effective - your Asterisk server would be aware of all of the volunteers' Skype accounts, and could route the call to whoever is online at the time someone calls. If no-one is online, it could route the call to someone's mobile (to ensure someone always answers a call).
AsteriskNOW has a graphical interface which will let you set up all of these features without needing to maually write dial plans: http://www.asterisk.org/asterisknow/
My email is in my profile if you have any follow-up questions, or would like more detail. Good luck!
I also second the other advice on: Affiliating yourself with an institution, ensuring all volunteers are suitably trained, and investigating any possible grants/funding from NGOs.
