I am not quite sure what you are asking.
The petition doesn’t care about the domain, though if you have any domain specific options they will have to be updated e.g. the URL after confirming email address.
If you are doing a plain WordPress export, I don’t think any plugins or their data is included. I use All In One Migration, but the current versions have a cost. There are others you can use to do an export e.g. duplicator. I don’t know pricing.
The best way to do it is as I described. Export the 2 database files using something like phpMyAdmin. Install SpeakOut! on your new WP site. Then use phpMyAdmin on your new site to first empty the dk_speakout_petitions
and dk_speakout_signatures
tables then import the old tables into the new empty ones.