There are actually two issues (maybe three) when buying non-food products.
1. is there going to be any animal products in the product? (vegan)
2. was the product tested on animals? (cruelty-free)
3. is the product ethically sourced and manufactured? (ethical)
In some cases, it's very hard to meet all three requirements without looking at something very expensive. In my mind, at that point, the "practical" part of the vegan definition comes up.
A bottle of shampoo or a pair of shoes might cost twice as much but that is within my comfort level. Those things last a long time and if you do a little "pro-rating" in your head it becomes inconsequential. You pretty much have to set your own threshold.
As far as animal products on the ingredients go, I hope maybe some other forum member can point you to a list of things to look out for. I can never remember those things. I have an app on my phone that is called "IsItVegan?" and when in doubt I use that. It's not perfect, tho.
The other thing to look for it one of the stickers that are on labels. There is the "certified vegan" sticker. There is also a number of organizations that certify cruelty free. Leaping Bunny, Cruelty-Free Kitty, and Bunny Free are the best known examples. I think they all have apps now.
Plus if you are at home you can use any of the databases. This one from PETA is pretty good
http://features.peta.org/cruelty-free-company-search/index.aspx
I don't think finding a good vegan computer chair will be any problem at all. A lot of the low-end chairs are all synthetic (i'm sitting on one right now). And even some of the high-end ones are.
Check out Polly and Bark's website. And search for "vegan".