IE11 - does a polyfill / script exist for CSS variables?

Yes, so long as you're processing root-level custom properties (IE9+).

• GitHub: https://github.com/jhildenbiddle/css-vars-ponyfill
• NPM: https://www.npmjs.com/package/css-vars-ponyfill
• Demo: https://codepen.io/jhildenbiddle/pen/ZxYJrR

From the README:

Features

• Client-side transformation of CSS custom properties to static values
• Live updates of runtime values in both modern and legacy browsers
• Transforms <link>, <style>, and @import CSS
• Transforms relative url() paths to absolute URLs
• Supports chained and nested var() functions
• Supports var() function fallback values
• Supports web components / shadow DOM CSS
• Watch mode auto-updates on <link> and <style> changes
• UMD and ES6 module available
• TypeScript definitions included
• Lightweight (6k min+gzip) and dependency-free

Limitations

• Custom property support is limited to :root and :host declarations
• The use of var() is limited to property values (per W3C specification)

Here are a few examples of what the library can handle:

Root-level custom properties

:root {
    --a: red;
}

p {
    color: var(--a);
}

Chained custom properties

:root {
    --a: var(--b);
    --b: var(--c);
    --c: red;
}

p {
    color: var(--a);
}

Nested custom properties

:root {
    --a: 1em;
    --b: 2;
}

p {
    font-size: calc(var(--a) * var(--b));
}

Fallback values

p {
    font-size: var(--a, 1rem);
    color: var(--b, var(--c, var(--d, red))); 
}

Transforms <link>, <style>, and @import CSS

<link rel="stylesheet" href="/absolute/path/to/style.css">
<link rel="stylesheet" href="../relative/path/to/style.css">

<style>
    @import "/absolute/path/to/style.css";
    @import "../relative/path/to/style.css";
</style>

Transforms web components / shadow DOM

<custom-element>
  #shadow-root
    <style>
      .my-custom-element {
        color: var(--test-color);
      }
    </style>
    <div class="my-custom-element">Hello.</div>
</custom-element>

For the sake of completeness: w3c specs

Hope this helps.

(Shameless self-promotion: Check)

This polyfill enables almost complete support for Custom Properties (not only root-level) in IE11:
https://github.com/nuxodin/ie11CustomProperties

How it works

The script makes use of the fact that IE has minimal custom properties support where properties can be defined and read out with the cascade in mind.
.myEl {-ie-test:'aaa'} // only one dash allowed! "-"
then read it in javascript:
getComputedStyle( querySelector('.myEl') )['-ie-test']

Features from the README:

• handles dynamic added html-content
• handles dynamic added <style>, <link>-elements
• chaining --bar:var(--foo)
• fallback var(--color, blue)
• :focus, :target, :hover
• js-integration:
  • style.setProperty('--x','y')
  • style.getPropertyValue('--x')
  • getComputedStyle(el).getPropertyValue('--inherited')
• Inline styles: <div ie-style="--color:blue"...
• cascade works
• inheritance works
• under 3k (min+gzip) and dependency-free

Demo:

https://rawcdn.githack.com/nuxodin/ie11CustomProperties/b851ec2b6b8e336a78857b570d9c12a8526c9a91/test.html