I am new to salesforce ecosystem and I am trying to make my first appExchange app but I am wondering if it is possible to use an external package like lwc-router while developing an application for appExchange? Any help would be appreciated.
Yes, you can. Just make sure that code that you use is licensed for you to use for commercial purposes. In this case, lwcRouter uses the MIT License, which means you can use it in your projects if you want to.