If you're using the Let's Encrypt client, you're probably using standalone mode right now, which does require you to stop the existing web server (because it provides an alternative/replacement for it for verification purposes). There are other modes that don't have this requirement, such as webroot and manual modes. If one of those modes works for you, it might address your concern.
There are also other clients out there
Some of them will also not require you to shut down your existing server, depending on the verification method used.