out "<form... <table...";
out "<input type=hidden name=\"counter=\" value=\"$counter\">"; # Den Wert von counter gibst Du einfach weiter
for (my $i=0; $i<$counter; $i++)
{
 out "<tr><td><input type=\"text\" name=\"Name_$i=\"></td>";
 out "<td><input type=\"text\" name=\"Vorname_$i=\"></td>";

out "</form></table>...";

for (my $i=0; $i<$counter; $i++)
{
 put ["name", "$name_$i", "vorname", "$vorname_$i"...], "DeineDB";
}