Dell ONLY supports Vista 32 bit on this model, however in most cases the Vista drivers all work in Windows 7 (as long as you have Windows 7 32 bit).
On a new install the correct and required install sequence on a Dell is (1) Install OS (2) Install Dell Laptop Support Software (on applicable models) (3) Install motherboard chipset drivers (4) Install Device Drivers including sound, video, ethernet, etc.
The motherbord chipset drivers are key to a successful install and there are two chipset drivers listed on the Dell downloads - and Intel and a Ricoh. One of the main things the chipset drivers do is define (identify) all the devices on the motherboard. If you skipped the chipset drivers, install them and then the sound and anything else that is missing.
Here is the Dell driver downloads for your PC Drivers & Downloads