I recently received tokens from my first ever bounty campaign and I was wondering if it's possible to transfer tokens from my MEW account that doesn't have any ETH yet to an exchange site like Binance?
Not possible mate, sending tokens to any exchanges or transferring to other wallets requires gas to execute any transactions. Gas fee is deducted to the amount of eth you hold in your Mew wallet. You need to fund your eth wallet because Mew wallet transactions are not free.