Scala Web applications do not typically use the Servlet API directly. If this is done, it is done solely by the intention of the developer, and Scala is used just like any other Jvm language. Nothing prevents you from writing servlets in Clojure or Groovy. Use the language as you would use Java.
However, in the world of Scala it is generally accepted not to use application servers. Primarily due to the fact that they complicate the "deployment" (deployment) of the application, as well as slow down the development process. Of course, there are frameworks that rely on Servlet Api, but access to servlets there is deeply hidden from the developer. Pay your attention to Scalatra as far as I know it supports the ability to work with application servers and servlets, but not directly.
What is accepted in the world of Scala? Now the de facto standard is Play framerowk . You can download and install it as part of the Activator platform.
An activator is an application that contains a number of ready-made templates for specific types of applications: web applications, SWING applications, and others. Install the typesafe template you are interested in, download it and use it.
IntelliJ IDEA can easily pick up any sbt / gradle / maven project (and the activator only offers templates using sbt) if, of course, Scala plugin is pre-installed for it. (Scala plguin for IntelliJ IDEA includes sbt support). Support for the Play framework is only available in the Ultimate version of IntelliJ IDEA. Scala IDE has free support for the Play Framework, I hope this information will help you