Is it secure to collect common public addresses under a wallet using BlockCypher Wallet API
When using a third party, you will be trusting them to give you correct information. This is not just about being dishonest but also about not having bugs. 

For instance consider the recent cve-2010-5141 vulnerability. If you are running a full node on your own, you can know what is going on and you can upgrade when you find out about it. But if you rely on a third party you have to trust that they upgrade and are aware of it.

As a matter of fact they apparently have not yet upgraded and on TestNet they are showing a block with exploiting transaction as valid. So theoretically you, as an exchange, can lose money if you were relying on them and if this was MainNet.