You can file a claim immediately after getting renters insurance, as long as you scheduled your policy to start immediately.
Your insurer will ask you when you want to schedule your policy to start when you buy it. If you’re currently uninsured (i.e. not covered by an existing renters insurance or homeowners insurance policy), you should schedule it to start right away in case there’s an accident later that day and you do need to file a claim immediately.
Note that you can only file claims for incidents that happened after you scheduled your coverage to begin. This means that if you buy a policy and schedule it to start immediately, and then later that day there’s a fire in your kitchen, you can file a claim for it. However, if you buy a policy and schedule it to start immediately, you can’t file a claim for a kitchen fire that occurred three hours before you bought the policy.