Is there a Mostly Cotton or Linen Travel Vest with hidden inside pockets?

Check out this site. Not cotton, but very happy with the three I own and never give the fabric a second thought. Some inside pockets (one with a zipper) but not so many that you stuff them and look lumpy.

https://www.magellans.com/shopcategory.aspx?ID=114,5497&srsltid=AfmBOoquGgVVd3eaHkdDG4iqPXzxfaiw2vvLNmJhCFPZebreNjoLE2o9

Rather than a vest, you might consider a shirt. Lots of companies make them with inside or accessible zipper pockets, including Scottevest.

We've perhaps a dozen travel (aka photo) vests. At least three (includes Domke, Orvis and Banana Republic) are 100% cotton (and the reason we proceeded to purchase the others which are nylon). We refer to our cotton vests as "triple-H": Heavy, Hot, Hydrophilic. The latter can be mitigated by periodically treating with water-repellent spray but tends to make them even hotter. Nylon vests are lighter, dry quickly and despite being less breathable tend to actually be less insulating.

Paul is right, and you could always take the sleeves out of a slightly bigger shirt if you really want a vest look

I have several of the Orvis river guide long sleeve shirts,deep side pocket with a zipper. Cotton,very well made and classy. Not cheap but I had mine for several years.

There is a current travel vest conversation under packing, fyi

There used to be a great one. Willis & Geiger. Look at the vintage sites.