I only need a copy of the blockchain to accept Bitcoins from others. I don't need it to spend them.
Is there a wallet app for Android that doesn't require me to download the blockchain?
There's no app that allows you to download the blockchain. Mostly due to the size constrains. A few good ones are Mycelium, Bitcoin Wallet (Download the one with Bitcoin Wallet developers as the developer]. I stongly advice against using anything by blockchain.info.