What is the allowable distance that a vent pipe can be from a fixture.
Depends in size of pipe trap size and vent size. Ex: 2 inch trap minimum 1-1/2 vent pipe is 5 ft max critical distance
We are looking at a bathroom
4 inch drain for toilet dumps to a 4 inch main drain top vent 4 inch through roof
1-1/2 drain sink feeds into a 2 in drain y’s into the toilet drain.
Toilet and sink at 6.5 feet from the main drain.
How do they drain? 
4 inch drain is 10 ft max and the 1-1/2 into the 2 inch drain would be a 2 inch fixture drain at 5 ft max
BTW. The distances i gave are from a vent pipe not the stack. Your sink may be fine if there is a vent pipe between the sink and 4 inch main stack.
This is installed much like the older homes used to be
a single 4 inch pipe from basement to roof all drains go to this pipe. smallest drain trap on the system is 2 inch with the fixture being around 6 1/2 feet from the drain/vent. system is considered wet vent.
the chart I was looking at earlier showed an allowance of 8 feet for that drain.