That's a tough one, because bad vocals are generally a turnoff for me, so I tend to not pay much attention to bands with bad vocals.
Megadeth is probably the most obvious example I can come up with. Mustaine's style is distinctive and fits the music. But I don't think anyone would accuse him of being a good singer, Mustaine himself included.
Flying Colors also comes to mind, I guess. Casey isn't really a "bad" vocalist, but just isn't my thing.
Depending on what you consider "weak," I guess an argument could also be made for Floyd. Neither Gilmour nor Waters were technically great singers. But the thing is, despite neither of them maybe having technical brilliance or being flashy, the used what they had very well, and it fit the music and conveyed a lot of emotion. Kind of like a Gilmour guitar solo--you may only have three notes used for the entire thing, but lots of emotion conveyed with those three notes.