The NPC market, which sells things from dead characters, can be accessed from any trade station or commerce center owned by a trading faction. Access is very spotty due to the circumstances of the listings.
There is a small store available for new players on most homeworlds, which sells a blaster, knife, and staff.
However the greater majority of players purchase and obtain weapons through buying them from other characters or weapon production factions. The trading forum and IRC are the main outlets for this method.
There are two main types of trading being discussed here: trading in the Galactic Marketplace, and trading with other characters.
The Galactic Marketplace is part of the Darkness interface, and can be accessed through the 'Market' link in the Members Menu on the right side of your page. You can view what is currently for sale in the Marketplace at any time. However as the Rules say, "To buy from (or sell to) the market, you must be inside a Trading Faction owned market, which means inside a Commercial Centre facility or a Trading Station (1 or 2), or inside Centrepoint Space Station." Here you will be able to buy entities that other players have already sold to the Marketplace, entities of players that have died or entities from factions that have dissolved.
The other type of trading is where you trade directly with other players. This can be done by arrangement through DMs, IRC, faction forums or CPM which is kind of like SWC ebay.
That is the advantage of the NPC shops founding in starting cities for races; when you buy something there it is at your location. The item and weapon selection there is very limited though, but if you are just looking for something while you wait for a better item delivery that might be a good choice.
If you have already joined a faction, you could also talk to your faction leader about what you want to do and he may provide you with some equipment and transport for it.