Some printers will handle the thin paper with no problems & others have to be coerced. Also, the age & condition of the rollers that pickup the paper will have an effect. When using non-standard paper, it may be easier & less frustrating to use the manual feed or straight through paper path. Some printers just won't handle paper as flimsy as onion skin or vellum though. I started paper piecing with some very thin typing paper and loved it but had to feed it through one sheet at a time to print on it. Now I can't find typing paper so I use newsprint (tablet paper) that I found at a flea market but I've seen it at office supply stores also. Not quite as easy to remove as the typing paper was but still easier than copy paper.
When I was in a book club, the books came with some type of very thin packing paper--like gift wrap before color is added. It could be ironed out, trimmed to size and fed through the printer or traced through. It was a little heavier than vellum but thinner than the typing paper. Long story short--I think gift wrapping paper will work if you can find lighter colors or the source before the colors are put on it.