Quantity can do what you expect if we add an extra definition as follows:

Quantity[1,"Dollars"];
Unprotect[Quantity];
Quantity[x_,"Dollars"]+Quantity[y_,"Dollars"]^:=With[{s=x+y},(
Quantity[0.01(Round[100 x]+Round[100 y]),"Dollars"])/;Head[s]===Real];
UpValues[Quantity]=RotateRight[UpValues[Quantity],1];

Then we can do this:

a=Quantity[5306.47,"Dollars"];
b=Quantity[4,"Dollars"];
c=Quantity[1182.54,"Dollars"];
d=Quantity[50.26,"Dollars"];
e=Quantity[0,"Dollars"];
f=Quantity[377.11,"Dollars"];
g=c+d+e+f;
h=Quantity[3700.56,"Dollars"];
i=a+b-g-h
(*  $0.00  *)

However, if you evaluate b, it will be displayed as an integer number of dollars. If you want to see $4.00 instead, add the following definition:

Quantity[x_Integer,"Dollars"]:=Quantity[N@x,"Dollars"];

In[122]:= Clear[a, b, c, d, e, f, h, g]
a = 5306.47;
b = 4.;
c = 1182.54;
d = 50.26;
e = 0.;
f = 377.11;
h = 3700.56;
g = c + d + e + f

Out[130]= 1609.91

In[133]:= N[SetAccuracy[a + b - g - h, 2]]
Out[133]= 0.

In[134]:= N[a + b - g - h, 2]
Out[134]= 9.09495*10^-13

..? Thanks for the proposal, but this following code doesn't give the expected result (0) on my side ...

Clear[a, b, c, d, e, f, h]
a = 5306.47 
b = 4.
c = 1182.54 
d = 50.26 
e = 0.
f = 377.11 
g = c + d + e + f  // AccountingForm
h = 3700.56 
i = a + b - g - h // AccountingForm

Don't mix integers with reals, Rationalize[] is an ovekill because there is AccountingForm[], a formatter, with other words

Clear[a, b, c, d, e, f, h]
a = 5306.47;
b = 4.;
c = 1182.54;
d = 50.26;
e = 0.;
f = 377.11;
h = 3700.56`;

In[51]:= c + d + e + f // AccountingForm
Out[51]//AccountingForm= 1609.91

In[52]:= a + b - g - h // AccountingForm
Out[52]//AccountingForm= 0.

Here is a shorter way:

Machine precision numbers are not exact so there will always be rounding errors. This is not something specific to Wolfram language. If you want exact results, use exact numbers.

a = Rationalize@5306.47
b = 4
c = Rationalize@1182.54
d = Rationalize@50.26
e = 0
f = Rationalize@377.11
g = c + d + e + f
h = Rationalize@3700.56
i = a + b - g - h
(* 0 *)

For details, see this.