What's the cheapest way to achieve the salvaging monthly requirement?

I don't think this can be answered to your satisfaction without knowing more details (but this didn't fit in a comment, so I'll try to answer).

Is 120 "cheap" items (just search for armour or weapons with few if any filters and sort by price) in your budget if you don't play enough to get your monthlies done? Cheap is relative, but that's probably the quickest way of doing it.

On the other hand, it doesn't take that long to run around in a low-level area, especially where there are high respawn rates, and "tag'n'kill" maybe 150-200 things (however many you need to get the drops to reach your salvage target, you're bound to not get any drops on some or get "junk" drops on others, but mostly it should be blues and greens or at least white items you can salvage). But, again, "not that long" is also relative. An hour of online time should easily be enough. If you've explored a bit, you'll know which spots are good. Off the top of my head, I'd go to the camp in Brisban Wildlands, in the northern part of Hidden Lake - it gets attacked almost continuously by bandits that are easy to kill for an experienced player. Pretty sure I could get 120 drops there inside 20 minutes.

Also, having guildies help you out can be an enormous bonus. Join a nice guild if you haven't already, if a few of them send you a mail with 5 blue items each, you'll be done in no time. Or they can run around with you to kill stuff quicker.

Buy orders for most blue and green weapons/armors are below the price of the materials they salvage into (sell prices are just slightly above - any item that has its sell price dip below equilibrium is quickly snatched up). You can easily fulfill the achievement AND make a slight profit by doing this and reselling the materials (or you can keep them, up to you).

Of course, "easily" also depends on your patience in setting and resetting orders (there are others doing this) on dozens of items - you'll hate the TP UI if you do this.