what is the best way for deployment
The one you can afford.
is running your own server advisable.
Fine by me, although for cost saving purpose, I suggest running on Kubernetes, as you can have servers that run your app and only your app, no other things running in the container that add to your bill.
is there any licensing cost or fees involved for deploying a Lazarus application.
If you merely use standard components: nope, LGPL + static linking exception allows you to create commercial apps with 0 requirements to link back or pay. Otherwise, read other 3rd party libraries/packages you use, as they may have their own licenses.