(*)In Europe carriage suspension (on chains) is recorded in visual images and written accounts from the 14th century ('chars branlant' or rocking carriages), and was in widespread use by the 15th century.
(*)Julian Munby, ‘From Carriage to Coach: What Happened?’, in Robert Bork and Andrea Kann (eds) The art, science, and technology of medieval travel (Ashgate, 2008), pp.41–53.
Introduced to Europe during the Renaissance. The Renaissance and Reformation both reached Russia via Peter the Great's Grand Embassy 1697–1698.
See the Russian Foreign Policy in the Twenty-first Century and the Shadow of the Past. under the heading Idea's of Europe.. page 398
Wiki. Peter the Great's Grand Embassy. for dates.
So that's probable a fair guess as to when they got suspensions on their carriages.