Use a Carrier Sheet to Hold the Envelope in Place
Using a carrier sheet to hold your invitation envelope in place when going through the printer is one of the best ways to print invitation envelopes, and is pretty easy to do, although can depend on the thickness of your envelope. There are several ways of doing this.
Whichever method you choose, you will first need to play around with printing an address onto a regular sheet of paper to check the positioning. Once you are happy where the address is going to be printed, you can put your envelope onto the carrier sheet in the relevant place. Remember to use the same font when experimenting as you will be using for your envelopes.
The first method involves attaching your envelope to a regular sheet of paper. You could use a thinner sheet of paper if you can find it - it doesn't matter how cheap it is as this is just being used to hold your envelope in place. You will need to do this by using some low tack tape, and make doubly-sure that it is stuck down perfectly with no raised edges. If this gets stuck in your printer, then try carefully deconstructing the envelope and attaching it in its flattened out state, so reducing the thickness of the paper.
If you have many envelopes to print, then you might find purchasing an adhesive machine (such as Xyron), a worthwhile investment. You can buy cartridges with repositional adhesive, so you can just whiz your paper through the machine and it will be coated on one side with a sticky substance a bit like that of a post-it note. This has the advantage of holding your envelope in place perfectly, although you will need to make sure that all other sticky areas of the carrier sheet are covered with paper too, or your printer will start to get a sticky build-up.
You could also try tracking down some repositional adhesive spray, available from craft stores, but be sure to spray the relevant area of the carrier sheet rather than the envelope.