Okay, after much trial and error I have figured it out.
I had to use a combination of URL encoding and HTML encoding.
In the end, there were 3 symbols the software/browser/server was having trouble with: '&', '{', and '}'
For the '&', HTML encoding was needed.
For the '{', HTML or URL encoding was accepted.
For the '}', URL encoding was needed.
In order to get the hyperlink to work I had to delete everything and then go back and enter the new url into the system.
The end encoded result: http://www.mycompany.net/records/personal/details.asp?taxyear=2015& ;ppt=A& ;acct=%7Bacct%7D
I added the spaces between amp ; so this site would not automatically translate it to '&'. The actual encoded url does not contain those spaces.
And after all of that, I needed to submit the hyperlink, click ok for the widget, then go back into the edit mode again and once more click ok. If I only did the process once, then my hyperlinked field did not translate to the actual account. For some reason, going through the process twice jumpstarted the link and my {acct} was linked to the number.
Thank you to all who helped and Vince, Thomas B, user30184 and Mintx in particular for steering me in the correct direction.
&
(& followed by a semi-colon) and not by a colon, correct?