As DOGE is such a new asset, it's yet to be listed on major exchanges. You can still purchase DOGE using a DEX (decentralised exchange) though, which just means there are a few extra steps. To buy DOGE right now, follow these steps:
3. Connect your wallet to the 1INCH DEX
Head to 1Inch, and 'connect' your wallet to it.
4. You can now swap your BNB for DOGE
Now that you're connected, you'll be able to swap for 100s of coins including DOGE.