Add linting to frontend (#4341)

# Description of Changes
There's no current linter running over our TypeScript code, which means
we've got a bunch of dead code and other code smells around with nothing
notifying us. This PR adds ESLint with the typescript-eslint plugin and
enables the recommended settings as a starting point for us.

I've disabled all of the failing rules for the scope of this PR, just to
get linting running without causing a massive diff. I'll follow up with
future PRs that enable the failing rules one by one.

Also updates our version of TypeScript, which introduces a new type
error in the code (which I've had to fix)

.vscode/extensions.json

@@ -19,5 +19,6 @@
     "yzhang.markdown-all-in-one", // Markdown All-in-One extension for enhanced Markdown editing
     "stylelint.vscode-stylelint", // Stylelint extension for CSS and SCSS linting
     "redhat.vscode-yaml", // YAML extension for Visual Studio Code
+    "dbaeumer.vscode-eslint", // ESLint extension for TypeScript linting
   ]
 }