Will upgrading to 4GB RAM on a white MacBook work?
All white Macbooks can certainly be upgraded to 4 and -in some cases- 6GB of RAM.

Depending upon your model, you should check OWC's Memory Upgrades and go from there. I believe other vendors (Crucial, Newegg, etc.) will have similar pages with similar info (and similar prices). I've brought different memory upgrades from OWC and never had a single failure so far. 

I have an old Macbook 4.1 White Core 2 Duo 2.4 and can go up to 6GB. So best thing to do is check your exact Macbook model and see what they offer.