A passive tracer is released for a long time at a constant rate R at a
height z = H above the ground, where the average wind velocity is S
and is in the x direction. The release point is within the boundary
layer of the atmosphere. What is the concentration of the tracer at
the ground (z = 0) as a function of x and y?

Sorry - I'm a bit confused here.
First, you have defined x to be the wind direction, but what is y?
If x and y refer to the position on the ground plane, as z refers to
the height above it, the you need to specify the wind direction as a
vector - and please, vall it something other than x :-)
Second, it seems to me that the ground concentration will also depend
on S, not just x and y.

Can you straighten me out here, please?

x, y, and z are a rectangular coordinate system.  The average wind
speed is S, a constant, and is in the x direction.  Anticipating
another question along these lines, there is no time dependence in the
problem. The answer I'm looking for is the time-averaged concentration
at the ground.