So I see 3 solutions if your PaperWhite firmware is lower than 5.6.x and you have some spare time to JB it.
Here are 3 solutions:
0) First use JailBreak
https://www.mobileread.com/forums/sho...d.php?t=186645 and KUAL
https://www.mobileread.com/forums/sho...d.php?t=203326 (you'll need it for any of these applications)
1) Try some other reader application to use interactive pdf to fill up the forms:
KOreader
https://www.mobileread.com/forums/sho...d.php?t=209276
Hacked Up Reader
https://www.mobileread.com/forums/sho...3&postcount=13
2) Try other web-browser for offline forms usage
Skipstone
https://www.mobileread.com/forums/sho...50&postcount=8
Midori
https://www.mobileread.com/forums/sho...d.php?t=210437
3) Try alternative means to in your case I have some suggestions:
The simple one is to use a LeafPad
https://www.mobileread.com/forums/sho...8&postcount=11 - text editor
The best one(I think) is KindleReminder from Kindle Menu pack
https://www.mobileread.com/forums/sho...d.php?t=225860 - simpe TODO tasker, but looks and feels like forms I don't know your needs but it's perfect in my opinion.
Also you can ask Aeris or any other people on development forum to tweak KindleReminder for changes you need in it, they might just help.
P.S. List of most of KUAL applications
https://www.mobileread.com/forums/sho...d.php?t=205064 maybe you'll find something else ))
P.P.S. Also maybe it's worth mentioning Gargoyle (IF player)
https://www.mobileread.com/forums/sho...d.php?t=223455 it's like step-by-step linear question/answer/reaction player, it can be used to remember choices, or even show results like in psychological tests, but I know nothing about all "Supported formats" or how these "games" can be created.