You certainly can. You can do anything that works. Melville does something very like this at one point in Moby Dick, so there is good precedent for it.
The thing is, why are you doing it? Why break convention? Any time you break convention, you call attention to what you are doing. When you follow convention, you text tends to become transparent and the reader simply sees that scene you are creating. When you break convention, you call attention back to the text itself.
There are certainly writers who break convention. Cormac McCarthy does not use quotation marks. I have no idea why, but I notice that fact that he does this, which makes me more conscious of the text. This might be why he does it, since his text is very deliberately poetic. (Again, it may simply be because he is Cormac McCarthy and he can do what he likes.)
Using a script format like this seems to add a kind of staccato tone to the dialog, as if it is spit back and forth rapidly. But that's how it strikes me. It may strike others differently. Being unconventional, it does not come with any guarantee of how different readers will interpret it.
In the end, though, as an author you are entitled to literary innovation. It may be worth it if it produces some worthwhile effect for your intended audience. But be aware that it will almost certainly rub some people the wrong way.