There is not much information about this vehicle. I found a few russian articles about it. Here a rough summery:
In March 1942 soviet army found an ammunition stock with German 28 cm Wurfkörper Spreng and 32 cm Wurfkörper Flamm rockets in the village Konduja near Leningrad. The rockets were send to Leningrad where the soviet engineers copied them and made their own rockets: M-28 (MTW-280) and M-32 (MTW-320). MTW stand for мина тяжелая, вращающаяся - heavy mine, rotating.
The LAP-7 was designed/released in autumn 1942 and is based on an modified(?) GAZ-AA chassis. LAP stands for Ленинградский артиллерийский полигон - Leningrad artillery range.
The six launchers were paired up and the angle of every pair could be adjusted separately. That means that it was possible to hit three different targets with one salvo.
The LAP-7 played an important part in defence of Leningrad and later on also in the break through of the Leningrade blockade. The Leningrad Front was the only front where this vehicles were used.
Unfortunately there is no infomation how many of this vehicles were build.
Hope this is enough information for you.
9. August, 22:24