Where is a reputable place to download Ultimaker Cura? I noticed that my version of Ultimaker Cura was out of date, so I tried to go to their website and download a new version.  For whatever reason, I could not get the download link to work properly in Firefox with a slew of security extensions. 

Where are reputable mirrors for Cura?

Download.Cnet.com and Sourceforge came up as sources, but years ago they both went to the darkside (SourceForge: HowToGeek, thecomputerpeeps).  I don't see anything else that looks even remotely reputable in the search results.

I did some analysis, and it appears that the drop down and form don't work correctly if you select "I don't want to share any personal information", leading me to think this is web page bug.
The answer to your question is: &quot;From the application developer itself!&quot;.
Below is the explanation how you can (directly) download Ultimaker Cura from the most reputable source: Ultimaker.com

Regular download
When clicking the link to download of the Cura application download button from the Ultimaker website you are presented with a pop-up GUI that asks you if you want to share where you are using the program for. When you select the required field and press download, the download starts. If this doesn't work, a direct download approach can be used.
Direct download
If the regular download method fails, or you do not get the pop-up GUI, you can use the direct link for Ultimaker Cura from the developer's website:
https://download.ultimaker.com/cura/Ultimaker_Cura-4.1.0-win64.exe
This link can be found at the download directory at the website:
https://download.ultimaker.com/current/
This web directory overview also shows all previous releases and installers for other platforms.
As of version 3.6.0, the download does not appear to be in the list, this link will show an overview: https://ultimaker.com/en/products/ultimaker-cura-software/list
Do note that the latter link will show the pop-up.
As of version 4.1, the web directory overview link will serve a 404.
Github
The Ultimaker Cura Project has a Github, where they offer the current and previous editions under:
https://github.com/Ultimaker/Cura/tags