The production of safety belts in Soviet Union was started in Estonian SSR by NPO Norma which still exists.
According to AS Norma website:
Our connections with the international automotive industry were established in 1973, the year in which Norma started producing safety belts for cars.
The obligation of using safety belts (if a car is equipped with them) was introduced in the Amendments to Traffic Regulation Orders which came into force on 1st April 1975.
Here is March 1975 issue of Soviet magazine "За рулём" (At the wheel) which discusses new regulations.
BTW. Driving an old car without safety belts equipped is legal even nowadays.