I've seen browser based searches but there's no way the project itself would support a full browser version. No one would bother running the client so the whole thing would seize up.
People who are invested in the platform -- OB1 and its investors, vendors who depend on the platform for sales, maybe even a few buyers who want to support the network -- would run the client.
I'm not sure exactly how OpenBazaar's P2P networking works but in theory, a lite browser-based version that pings nodes should be possible. They should consider something to make it more accessible.
There is browser version! Though it does not yet support buying, that functionality is in development now. The goal here is to eventually have OpenBazaar completely browser-based. https://openbazaar.com
Also newly released is a mobile experience for OpenBazaar called Haven: https://gethaven.app
Happy to answer any questions anyone has about the platform & various clients, I am the Marketing & Communications Lead at OB!