Sorry to hear about your iPod touch, hopefully I can help get it repaired.
To start off, yes, Apple has a page where you can determine your device's Apple Care Plus eligibility. That can be found here: https://checkcoverage.apple.com/. If you have any trouble finding your serial number, go ahead and visit https://support.apple.com/en-us/HT20157/.
Since the vast majority of US Apple Stores are still closed as of now, your best bet is to start a repair claim via Apple's website: https://support.apple.com/repair. Once there, select "Start your repair request now," choose "More," then click iPod, and finally iPod touch. In the search bar, simply type in "battery" and the second option should be "Battery requires service." At this point you can choose to go ahead and send in your iPod for repairs, talk to an Apple representative via chat or phone call, or schedule a call for later if you are currently busy. If you choose to send it in, make sure you have your serial number handy, as Apple needs that to finish the repair claim.
Hope it all works out!