No. An H4 visa is a dependent's visa and, as such, the holder has no recourse to public funds, no social security number and is not permitted to earn money while in the US. This includes working on the internet. You are allowed to do voluntary, unpaid work for non-profits etc. Be aware that it is not permissible to "volunteer" for a job that would normally be a paid position for an American.
You can, however, get a driver's license and attend an educational institution (full or part time). Although an H4 holder may not receive a Social Security Number (SSN), you may apply for an Individual Taxpayer Identification Number (ITIN) for use when filing your yearly tax return, and this number may be accepted in some situations in place of an SSN.