How do I go about removing a content type such as webforms from the site search results?
You need Search configuration module.
It will allow you to exclude a content type from search results and even particular nodes.
Drupal's contrib module Search Restrict is a lightweight module that just does the basics - it allows administrators to restrict by role who can search for each content type. It may be a better option than Search configuration in that it only does what you're asking for - restrict content types from search results based on a user's permissions. If you're looking for something more configurable, Search configuration may be the way to go.
From the module's page:
Approach of this module is to re-write the search query, so that content is indexed and available as search results to users in role(s) that have permissions to view it, but not displayed to other roles. Integrates with Apache Solr to exclude results based on the role of the person searching.
Simply download, install, and enable the module. Then to restrict (by role) which users can search which content types, edit each Content Type individually and there will now be a 'Search Restrict' section option (below the Display Settings). It allows you to specific, for each content type, which roles can search nodes of that content type.