Well, three ways immediately come to mind.
First off, you could try making a link to the form using ".../form.php?textarea=".urlencode("a=1\nb=2\nc=3\na + b - c"); or something similiar. If it fills it in the textbox, you have a winner; just depends on how the form is coded.
Second, you could try imitating the post on your site, i.e. copying their form code and changing the relative reference in ACTION="" to an absolute one, i.e. ACTION="form.php" becomes ACTION="http://remotesite.com/form.php
Third, and this would be my choice, because it involves text parsing (MAN I love text parsing!), you could create a script that takes input for that field on your site; on submit, it could open itself or another script in a new window which would grab that form page, parse it until it found the text field/textarea you want (<textarea name="textarea" cols="60" rows="5" wrap="VIRTUAL"></textarea>, in the example you sent), and insert the input from your original form right between those start/end textarea tags.
[ edit: method 3 would also entail changing the relative references to absolute, as the page still ends up on your site ]