Like, don't bring too much stuff
On the contrary. Bring your companions with you, have them carry as much ammo as they can, take it from them, THEN activate the satellite after dismissing them.
Many enemies in the Big Empty have ~700 health and 20 DT, meaning they eat bullets like candy.
Bring money too. You may have read the above line and thought "then I'll bring a good melee weapon/fist." Those help (Paladin Grill is a good choice, a useful melee weapon will be provided for you), but again, they'll break a fair amount since the enemies demand so many hits, so you'll want cash to pay for repair costs.
Might also consider bringing Lucky specifically, if your luck and agility are good enough to use it properly. That gun saved my ass.
For the record, my first playthrough of OWB (blind), by the time I was done, I only had ammo left for Lucky (.357 magnum is one of the few ammo types actually found in OWB) and a couple shots for my Brush Gun, though I only had those because I was specifically conserving them for the possibility of tough fights. I went in with 5mm (2000+?), .357 (300-400), .45-70 Gov't (600-800), .308 (500-600) and a fist, then had only .357 (and my fist obviously) by the time I left, really.
Had other characters go in (not-blind, I knew of the ammo issues) with loads of 12.7mm, .45 auto, etc, and it's always the same. Last time I even brought 4k 12.7mm bullets, spent them all. .357 and .44 magnum are the only ammo types that have an advantage.
Just be careful with your ammo. Bring a sidearm perhaps and, before each fight, ask yourself "do I REALLY need to spend rifle ammo on this." If not, use a pistol. Or a fist, if you have enough caps to keep repairing said fist. Weapon repair kits might also be a good idea, since, if you're wearing anything but Heavy armor, you'll need that money to repair your armor.