dc, 16 11 bytes:

?d12742+*vp

Prompts for input through command line in kilometers and then outputs distance in kilometers.

Explanation

?           # Prompt for input
 d          # Duplicate the top-of-stack value
  12742     # Push 12,742 onto the stack
       +    # Pop the top 2 values of the stack and push their sum
        *   # Pop top 2 values of the stack and push their product
         v  # Pop the remaining value and push the square root
          p # Output the result to STDOUT

This takes advantage of the following:

((6371+h)**2-6371**2)**.5 
=> ((6371**2+12742h+h**2)-6371**2)**0.5 
=> (h**2+12742h)**0.5 
=> (h*(h+12742))**0.5

jq, 18 characters

(12742e3+.)*.|sqrt

Yet another copy of the same formula.

Sample run:

bash-4.3$ jq '(12742e3+.)*.|sqrt' <<< 11.5
12105.087040166212

On-line test

Haskell, 22 bytes

d h=sqrt$h*(h+12742e3)

Usage:

Prelude> d 11.5
12105.087040166212

Pointfree: (23 bytes)

sqrt.((*)=<<(+12742e3))

05AB1E, 13 12 10 bytes

Saved 2 bytes thanks to Emigna.

Since there is no trigonometric functions to be called using OP's assumption that the earth is locally a plane, it becomes possible to make an 05AB1E solution.

•1#oC•+¹*t

           For understanding purposes input is referred to as 'h'
•1#oC•     Push 12742000 [ = 2*R, where R is the radius of earth]
      +    Compute 2*R+h
       ¹*  Multiply by h. Result is (2*R*+h)*h
         t Take the square root of it and implicitly display it

Try it online!

R, 29 bytes

h=scan();sqrt(h^2+2*h*6371e3)

Takes input from stdin

Python, 34 26 bytes:

(-8 bytes thanks to Osable!)

lambda i:(i*(i+12742))**.5

An anonymous lambda function. Takes input in kilometers and outputs in kilometers. Invoke as print(<Function Name>(<Input>)).

PHP, 34 bytes

<?=sqrt((12742e3+$h=$argv[1])*$h);

breakdown

   r^2+x^2=(r+h)^2      # Pythagoras base
   r^2+x^2=r^2+2rh+h^2  # right term distributed
       x^2=    2rh+h^2  # -r^2
       x =sqrt(2rh+h^2) # sqrt

so far, this is identical to the old Mathematica answer

sqrt(2*6371e3*$h+$h*$h) # to PHP
sqrt(14742e3*$h+$h+$h)  # constants combined
sqrt((14742e3+$h)*$h)   # conjugation to save a byte
((14742e3+$h)*$h)**.5   # same size

now all that´s left to do is to add input =$argv[1] and output <?= - done

Mathematica, 16 bytes

Either of these works for both input and output in kilometres:

(12742#+#*#)^.5&

((12742+#)#)^.5&

This is a simple application of Pythagoras to the problem:

   x*x + R*R = (R+h)*(R+h)
=> x*x = (R+h)*(R+h) - R*R
       = R*R + 2*R*h + h*h - R*R
       = 2*R*h + h*h
=>   x = √(2*R*h + h*h)

Tcl, 49 bytes:

set A [get stdin];puts [expr ($A*($A+12742))**.5]

Well, I am brand new to Tcl, so any tips for golfing this down is highly appreciated. Like my other answers, prompts for command line input in kilometers and outputs in kilometers. Essentially a Tcl adaptation of my existing dc and python answers.