Do cells use active transport to maintain equilibrium with their environment?

no

Explanation:

cells maintain internal environmental equilibrium by the process osmosis.

when the cell is moves into a hypotonic solution, water molecules move into the cell by osmosis, hence creates excess water in the cell. In order to maintain water equilibrium, reverse osmosis occurs to move the water molecules out of the cell.

On the other hand when the cell is placed in a hypertonic solution, water molecules move from the cell to the outside environment, causing insufficient water in the cell, but to correct this, the cell again initiates reverse osmosis to bring water back into the cell.

Note: There is nothing like the cell moving from one solution to another to attain an internal environmental equilibrium.