I like the idea of bottom-decking cards, but this doesn't seem like a strong enough way to do it. If you draw this early in a shuffle, the benefit is extremely marginal. If you draw it late in a shuffle, it's comparable to a Royal Seal that offers no money at all, which, if you think about it, makes it a worse card than Copper.
The reaction component could be great in certain circumstances, but how often does someone trash something you might actually want to have? Most cards that go to the trash belong there. The exceptions: Bishop, Apprentice, Salvager, Trader. In most games, those cards aren't present -- and when they are, the presence of Scavengers on the board may mean that optimal play is not to use them anyway.
You could make a case for adding the Remodel family to this list, but if you're willing to gain the cards your opponent is replacing with better cards, that's a sign you're already losing. And Lookout might cause Provinces to get trashed, but it doesn't happen often enough to buy a card on the off-chance you could turn a freak event that will greatly benefit you into a freak event that will greatly benefit you more.
So I'm not sure how to make the reaction portion work. The action portion would be cool if you grafted it onto a stronger card. For example, this would be a harmless pick-up with extra buys (and combo nicely with Pearl Diver, which usually gets bought under the same conditions):
Scavengers
$2 - Action
+1 Card
+1 Action
--
While this is in play, when you gain a card, you may put it on the bottom of your deck.
And this, while a pretty weak $3 card, would probably not be too weak:
Scavengers
$3 - Action
+$2
--
While this is in play, when you gain a card, you may put it on the bottom of your deck.
This ought to work, too:
Scavenger's Village
$4 - Action
+1 Card
+2 Actions
--
While this is in play, when you gain a card, you may put it on the bottom of your deck.
Caveat: The non-terminal versions of these actually might be slightly too strong in multiples (because you could manufacture a power hand as you go) and might need a $1 price increase depending on how they playtest.