Can your broadband be shared with other devices aside from your laptop? If so, then yes the Roku will work. If not, then no. Roku requires an internet connection - either wired or wireless. The Roku makes the connection to the Internet, then brings your videos back to you and outputs to a TV screen via HDMI or composite video. The laptop is pretty much out of the equation unless you MUST use it to get your Verizon broadband working (which wouldn't make sense to me, but I don't use Verizon).
Also, only attempt this if you have 4G or really good 3G connection. Anything less than full 3G and you just don't have enough bandwidth for streaming video. Do you remember what dial up was like? it would be like that again.