To investigate whether the issue could be a content blocker, please check these possibilities:

(1) Built-in Tracking Protection feature (especially in private windows).

Do you see a shield icon toward the left end of the toolbar when this problem hits? More info on how to use the shield to manage blocking: What happened to Tracking Protection?.

(2) Content blocking extensions such as Adblock Plus, Blur, Disconnect, Ghostery, NoScript, Privacy Badger, uBlock Origin or uMatrix.

These typically provide toolbar buttons to manage blocked content in a page. If you make an exception for the currently displayed site, does that help at all?

Neither of those seemed to be the issue. The captcha box was still blank.

What if you try the demo directly on Google's site (I'm assuming these are Google ReCAPTCHA's): https://www.google.com/recaptcha/api2/demo

The difference there is that the frame is "first party" instead of "third party".

The sample form did show up on that link.

Okay, do you block third party cookies, or Google cookies? This article will point you toward the critical settings: Websites say cookies are blocked - Unblock them.

That didn't seem to work either.

Make sure you are not blocking content.

Start Firefox in Safe Mode {web link} by holding down the <Shift> (Mac=Options) key, and then starting Firefox.

A small dialog should appear. Click Start In Safe Mode (not Refresh). Is the problem still there?

The captcha is still blank.

Start your Computer in safe mode with network support. Then start Firefox. Try Safe websites. Is the problem still there?

http://encyclopedia2.thefreedictionary.com/Linux+Safe+Mode Starting Any Computer In Safe Mode; Free Online Encyclopedia