Absolutely... I use to give it to pilots all the time from the tower. Although it's usually the ground or clearance-delivery position you'd get it from. When you call up, state that you'd like to depart VFR and request VFR flight following, and your destination. They may ask if you wish to continue FF with the Center if you're going into Center's airspace. This is because most approach control / tower airspace's have their own squawk code subset (such as 03xx or 04xx) for aircraft remaining within terminal airspace. If you plan to venture outside of their airspace, you'd need a Center code (like 4377, 2154, the dreaded 6666, etc).
You can always request / cancel flight following anytime while airborne as well, but it helps if you do it ahead of time. It's a lot easier for the ground controller to enter all the information in the computer ahead of time, rather than the busy approach controller having to change your information in their computer.
Pilots: I sure do recomend visiting any ATC facility to get a perspective from that side of things. You'd understand better what all they're having to do, and why they do certain things. Controllers are also just regular people, like you and I, so never be nervous and intimidated to talk to one. Some are grumpy, while others are more than happy to help in anyway. Some are moody (depends on the day), just depends! :-) Overall they're a great bunch of guys/gals on the other side of the mic.
Former FAA air traffic control specialist, pilot