Weapons and Equipment are Procure Off Site
Neither Blades in the Dark nor Scum and Villainy really take place in worlds where people derive large amounts of personal power from singular things they carry. Run down the list of organizations and look at their assets - very few will list anything like singular treasures. Warehouses full of ship weapons, streets full of gangs - power is projected from numbers and the will to command them. Even Lord Scurlock, the guy whose arcane delvings let him fight like thirty men, has got libraries of knowledge and racks of artifacts, and for all that he'll probably pull out some arcane thingus to trap half the crew behind a wall of ice, the operative power there is still Lord Scurlock's Lord Scurlockness.
Similarly, everything on the PC's sheet is not just the abstraction of some starting stash but is everything within their ability to replenish during any downtime, even if the last score ended with them tossed in an alleyway naked and bleeding. Everything is fair game for the GM to break because the PCs can get it all back - the Pilot's fine small urbot eats some heavy EMP and will need some TLC before it's useful again, the Spider's fine cover identity gets completely blown and they need to establish a new one.
The notes section of the character sheet is useful as an auxiliary inventory for all the things you acquire or craft during downtime - heck, the Leech is using some of it that way already.
You can also use it to track the stuff you pick up. I mean, you can totally pick stuff up. You've got arms, right? Manipulators? ...hokey religions and ancient telekinetic machines? The exact steps necessary to track it depend on what kind of stuff it is.
Stuff Any Crim Could Get - Load
Every piece of gear you bring along on a score has Load, even if that Load is "it has no Load". The same is true for everything you pick up, so respect your Load limits when you do - go up a category, or drop some stuff you "were" carrying to stay at your current category.
Why would you do this rather than just declare you have something? Sometimes you're just trying to replace something you lost. Other times there might be some small fictional advantage to picking up something right now. Still others this could be "normal gear", but not really something you'd think to load out with, for whatever reason.
Vex and Sector nail the engagement roll to sneak into the Scarlet Wolf outpost and decide they've done it by doing a masterful simultaneous cross-up takedown of the guards outside some neglected entrance. As they continue inside, Vex has an idea - why not loot the armor off the guards, to help them blend in?
The GM says sure, but it's still armor, so account for 2 Load, and you've got an improved position for trying to avoid detection since the Wolves are less likely to investigate someone wearing their armor.
Baz is the mechanic on a ship raiding a Hegemony salvage complex. A botched engagement roll means the quick zip into processing alongside a larger hunk of junk turned into a barely-survived crash that ripped a huge gash in the hull, and she asks the GM if she can scavenge some parts for repairs.
The GM says sure, there's generally serviceable wrecks all around, and gives Baz a choice - account for 3 Load now and continue on the score, and repair the ship at the end with improved position from studying the salvage in quiet moments, make the repairs now and spend an uncertain amount of time unable to help on the score, or hope there's time to make repairs when it's getaway time.
Specialty Stuff - Specialty Setup
Stuff that not any crim could get, like special gear from other playbooks or the stuff that might take an Acquire Asset action to take during downtime, is generally not offered by the GM to loot during a mission, just out of hand. Largely because, well, see the top. Most organizations, and most people within them, don't pose a threat just because of something they have that somebody else can take away.
But having something beneficial happen during the score is well within the realm of a flashback. Especially if it's "finding something useful on site", it might not be done by explicitly flashing back to the Acquire Asset downtime action, but some sort of flashback action can set up finding a more useful item.
Vex says they thought the the armor would just get them through the parts of the Scarlet Wolf compound that these guards could access, and they could make a break for their actual objective in their own time.
The GM thinks for a second, and says it wouldn't, unless somehow Vex and Sector knocked out two guards who were built exactly like them. They're welcome to set up the circumstances that led to those two guards being outside in a flashback, though.
Baz asks the GM if there's anything else she could find in the salvage yard. Being able to fit something like a Coherence Cannon to the ship would certainly help escaping, if it comes to it.
The GM says that while scrap is easy enough to find, actually finding an intact weapon system in everything the complex has salvaged would be almost impossible, unless Baz knew ahead of time where to look for one. She's welcome to say how she learned about where it was in a flashback.
Stuff You Want To Keep - Downtime
Regardless of whether or not you made the Acquire Asset roll as a flashback to get it, if it's feasible to hold onto some item of specialty gear from mission to mission, you'll need to make an Acquire Asset roll in order to do so. (Ordinary gear that just provided a positioning advantage because of the circumstance you found it has almost certainly lost that circumstance by the end of the score.)
You make the roll with a +1d advantage for already having the gear, but you must make an Acquire Asset roll to hold onto some piece of nonstandard specialty gear that you looted during a score - the same way you would for a piece of gear you already acquired, and for the same reasons. Actually adding something to your loadout on a permanent basis is more a matter for a long-term project that you undertake.
You can use the Acquire Asset rules for acquiring illegal items for the idea of "holding onto something someone doesn't want you to have". It'll rack up heat to model the additional cost of taking measures to secure it, and the heat will contribute to entanglements as normal.
Vex engages Red-Eye Romero, head of the Scarlet Wolf outpost, and lands a critical success on a Scrap. The GM says Vex knocks him out temporarily, instead of just being able to hold him off and break away cleanly. Vex says they want to lift Red-Eye Romero's urcloth bandanna as a trophy. The GM asks why and Vex says to provoke him into doing something stupid.
When Vex gets away with the bandanna (it's a 0-Load artifact that Vex doesn't know how to use) during downtime, the GM says that actually keeping it secure until the next score will need a tier-2 Acquire Asset success, to match the local resources the Scarlet Wolves can bring to bear, and the action will rack up 1 Heat - less than acquiring something the Hegemony thinks is illegal, but the Scarlet Wolves will certainly take notice. While the crew has the bandanna, they also have access to a 6-tick project clock called roughly "Make Red-Eye Romero Do Something Stupid".
As part of the harrowing escape from the salvage platform, Baz finds herself hotwiring a prototype Hegemony battle-walker and it survives all the way to the ship and back outside again. She asks the GM what it's going to take to hold onto this thing.
The GM says that she can use the Acquire Asset action to get supplies to repair and recharge it between missions, but they'll be full-on 2-heat illegal - and that bringing the walker itself on scores anywhere in populated space will start an 8-tick clock counting down to a Hegemony retrieval team arriving.
The GM also offers Baz two project clocks - a shorter-term one to understand the walker so that keeping it in repair won't involve buying flagrantly illegal supplies, and a longer-term one to really understand the walker so she can build her own version that the Hegemony won't automatically be after, though causing a battle walker's worth of mayhem on a score will still generally build heat.