Where does Bittrex/Binance (alternative cryptocurrency markets) get their coins?
Pure exchange like Bittrex neither need, or have use for a huge reserves of cryptocurrencies. Their service model is based on providing a platform for peers to sell and buy coins/tokens.
Also, they don't actually buy the coins from the sellers. Sellers/buyers are the one's who deposit coins to Bittrex's wallet, so Bittrex act like an intermediary between the seller and buyer.
There is an exception though, you can actually wire transfer to Bittrex and buy USDT, and they will purchase on your behalf from other sellers on a per-order basis.

Coinbase is different, they are more like a bank-hybrid with their own dedicated exchange(GDAX), and as a bank , they have a huge reserves of cryptocurrencies as they deal with buyers/sellers directly.