On Shabbat, is it permissible to seal an envelope that does not require moistening, ie an envelope that has a PEEL & SEAL flap?
Rambam Hilchos Shabbos 10:11 says that gluing two pieces of paper together on Shabbos is a Toldah (offshoot) of Tofer (the biblical prohibition against sewing on Shabbos).
המדבק ניירות או עורות בקולן של סופרים, וכיוצא בו--הרי זה תולדת תופר וחייב
One who glues pieces of paper or leather to each other with scribe's glue (קולן של סופרים), or anything similar -- this is an offshoot of תופר, and he is liable for it.
Therefore I would conclude, that no matter the method of gluing (whether regular glue, regular envelopes, or "peel & seal" envelopes), gluing paper together is forbidden on Shabbos.