Well the challenge with vellum is that the adhesive shows through. I know what a pain it can be to find adhesive that doesn't show just for scrapbooking or paper craft applications. (To the point that I usually just use brads)

They sell adhesives specifically for vellum at most scrapbooking stores, but I'm not sure how it would hold up to the weather. (worst case, the paper starts to curl up and you need to figure out another option in the Spring - it shouldn't do any damage to the glass that can't be fixed with a little goo gone and elbow grease)

If your heart isn't set on the vellum, I might check your local craft store for a product designed for glass.

Is the concern purely aesthetic? I don't think clear glass would look bad. Do you need privacy? What about etching the glass to make it frosted or painting the glass a dark but translucent color?

Here is a link that might be useful: craftfetish blog

Thanks craftfetish. I will look into those adhesives. I did end up calling 3M and they didn't have recommendations because the vellum is thin, they thought it would cause splotching. I thought of brads too but I think the paper would curl and they would drop out.
Getting rid of the old stainglass paper is for aesthetics. Putting up something new is for privacy. I don't want someone looking into the house when they knock on the door. I did consider painting the glass or using the stainglass in a bottle stuff they sell at the craft store. That is still an option. I've always been put off from etching because the chemicals are suppose to be so caustic.
I am considering the vellum right now because the effect is beyond my artistic ability if it came out the way I would hope - often things don't come out as planned for me.
I guess I will have to experiment here with a few options.

Maybe I can hide any spotching by antiquing the paper...and it might look better.

QVC sells something called wacky tac that is specially for putting vellum on paper and supposedly you can't see it. I am not sure if it would work for your project but maybe.

I am interested in this idea, too. I was thinking of using rice paper and tearing it into irregular pieces and ovelapping them in a "mosaic" pattern. Do you think this would work? And I haven't thought about the glue yet. Any ideas would be appreciated.

Mar cia, a bit late but thank you for the tip. I will look into it.
Ukigumo, I think torn vellum is effective. What is your substrate? Is it glass too or are you gluing onto something else? Do you care if it comes off clean again?
Talk about procrastination, I still haven't done this project but I did buy some acid free double-sided tape. I figure that is easily reversable if it doesn't come out the way I want it and I could go on to the next idea.

Loretta, I haven't had time to think this all through, but I was planning to glue the torn paper right onto the glass which is north facing, so no direct bright sunlight. I hadn't thought about cold.

I had wondered about using a diluted wallpaper type paste or craft glue that dries clear.

Now I am wondering about making panels -- such as one long panel of paper same size as window and gluing my random paper to that panel. Then I would attach the panel in the corners only, perhaps with doublesided clear tape.

Obviously, I haven't gotten around to this project yet -- if you have pursued it, I would love to hear your ideas.

I'm getting to it ... :-)
I picture wallpaper paste as getting crackled and yellow?

This would be more of a temporary fix, but if you haven't tackled this project yet it seems like a fun option...

One of the blogs I read shared this picture of her door, decorated for Valentine's. Since it is a temporary look, I'm thinking she used poster paint or something easy to wash off.

Here is a link that might be useful: Green Kitchen Painted Door

That is cute! She doesn't answer anyone about how easy it was to clean up though.