Speaking of space cowboys, like I was in the Lazy Laser post earlier today, here's Vant Galaide showing off a space cowboy outfit on Yavin IV. I deliberately made this one out of mostly adaptive vendor pieces, so anyone could get it.
There is one exception, though; the weapon I use (noted below) is from the cosmetic vendor, and costs half a million credits and 3,500 tech fragments, a high level currency. It's legacy bound, so once you buy it, anyone on your server that's one of your characters can use it, but it's not easy to get. The adaptive vendor pistol is terrible, so I don't recommend it, but you'll find tons of world drops on your first planet that will work quite well. And if you're playing—or have played—a bounty hunter, the Hunter's Blaster which drops near the end of the first planet and which is legacy bound is a great alternative, and is free to boot. However, you can't start with it.
Normally, of course, you can't get any of the adaptive gear until you go to the fleet, and normally you don't go to the fleet until you're done with your first planet, so that shouldn't be a problem, however. The craftable Custom-built Blaster Pistol, which you can equip at Level 20 or higher is another good alternative, as is the craftable Carbo-plas pistol which you can equip at level 10, or the Plastifiber that you can equip at level 16. They don't necessarily have a similar look, although they're mostly in the same size and chunkiness range, and because you can craft them pretty cheap at low level, they're great alternatives, and of course, since they're craftable, you don't actually have to buy them.
The pants on this outfit are also technically free, but they're hard to get, requiring grinding Heroics, being lucky in your RNG output, and generally unless you've somehow already got them, it's probably not worth going for. There I recommend some pants from either world drops or the adaptive vendor to replace them. The Sith Annihilator, TD-07A Blackguard and TD-07A Scorpion all offer interesting leggings options to replace the ones I used. They don't look the same, but I picked the ones I did because they're kind of unique.
Also, although the dye module I use is crafted and technically free if you have someone who can craft it, it's a high level dye module and you may not have access to anything like it. The Black and Light Gray is a suitable lower level alternative. But even then you may not have access to the color that you want. You could also buy the Primary Black Dye on the fleet, but it also requires special currency and isn't super cheap.
What are some other changes you could make while still staying out of cartel market items?
I like the Han Solo style vest, but if you wanted something else, the Republic side Scout with the same dye module would look pretty interesting. If you can find the schematic, smugglers can wear the old fashioned Drelliad jacket starting at level 11 or something like that. The Remnant Resurrected Smuggler is a great alternative if you have it, although like my pants above, it's hard to get if you don't have it already. Although at least its free. On the other hand, although not very cheap, some of the planetary pieces are at least easy to get if you have a character that can afford them, and they're legacy bound so you can easily pass them around. The Tarisian and Nar Shaddaa smugglers have a classic SWTOR look.
UPDATE: I almost forgot! Although the Republic Squadron Commander Pilot set is free for me, it may not be for everyone if you don't have access to the code that gave it to me. Which I can no longer find, so I presume it's no longer active. There is a belt that looks very similar in the Adaptive vendor, and of course, since the gloves cover up the bracers, it doesn't matter very much which ones you replace them with.
Also; not sure why I went to Yavin IV, actually. On second thought, it doesn't seem like a really cowboyish planet. But Tatooine... y'know, you get tired of doing everything there sometimes.