Yes, a renters insurance policy starts immediately after you purchase it, as long as you’ve indicated that you want it to start that very day. If you don’t need it to start immediately, you can also set it to activate on a future date, but usually not more than three months later.
If you’re in the process of moving (and you’re not able to transfer your existing renters insurance policy to your new address), it’s a good idea to buy a new one before you move. So don’t worry that you might accidentally pay for an extra policy on a rental you don’t yet live in — insurance companies make it easy to set the activation date, even if you’re purchasing online.