When dealing with a centralised exchange, there are only two points of interaction with the blockchain: Deposits and Withdrawals. In order to process deposits you will need to monitor transactions to certain addresses. A withdrawal can be processed offline with only the signed transaction being broadcast to the network. For safety reasons it is best that exchanges run their own nodes, as they can ensure their own data integrity particularly for deposits. It is less important for withdrawals as signing a transaction does not actually require a node, however accurate balance checking is obviously required in order to ensure that the expected amount of funds are held within an account.
2If you need to ask strangers on the internet how to run a business that deals with customer funds, I hope customers are smart enough to stay away. – Pieter Wuille – 2019-02-07T23:44:08.057