Hi, to buy NEO you don't necessarily need a wallet you can sign up to an exchange to convert your FIAT(normal money) into either Bitcoin or Ethereum and then NEO. You will need to go through a verification process before you can fund your account and then purchase NEO. I did not do this on Bittrex as I had Ethereum elsewhere and funded my account with that.

There are other exchanges that may suit you better you can have a look here: https://coinmarketcap.com/currencies/neo/#markets.

Something worth noting if you are buying NEO is to make sure you can earn GAS from it also. I know having it in the NEO endorsed wallets allows this and Binance also have this in place.

I hope this has helped in some way and good luck!