Of course you can add JWT to your existing PHP app, it's not "locked" to Laravel, far from it. Most of the frameworks that have these capabilities built in are just supersets of various components to make life easier for developers. You can of course implement these on your own in your application, given that you have the knowledge how to do so.
Since this is a bit abstract question without any provided examples, some answers with code snippets might just confuse you. Instead here are some resources where you can read more on the subject and get going: Sitepoint, JWT in PHP App and PHPClicks Token Based Authentication.
Another approach would be to craft a new Laravel project (or Lumen project if it's only a API and you don't want to render views) and import the current codebase from the other application to it.
If you want to make life easier for you, you can use some SAAS alternative for Authentication like Auth0.