Typically with a truly peer-to-peer network the nodes that are participating must be online in order to interact with each other. OpenBazaar is unique, however, because it is built on IPFS which allows a store to be accessible to buyers even when the store is offline. With IPFS, stores and listing content are cached by other OpenBazaar users, meaning if store content is cached, a purchase can be requested even when the seller is offline.
Payments behave a little differently if a store is offline when an order is received. Read more about Offline payments here.
Offline messages should be queued and pushed to a user when they log on next but with OpenBazaar still being a peer-to-peer network, messages are sent and received best when users are online.
Sellers are still encouraged to keep their stores open as much as possible but at the very least check in weekly. If a seller does not go online for 7 days their data may not be cached and they are unlikely to be visible to other nodes or show up in search.