Toilet venting, no stack installed
I suspect you have an air admittance valve. Based on your description perhaps it is under the sink or even (improperly) in the wall.

An air admittance valve is commonly used for venting things like islands where alternatives are complex and there is no stack. They often leak sewer gas, though, and are often prohibited for use.

On the bright side if you're going to replumb anyway you should be able to add a more reliable vent fairly easily (though perhaps a little tedious)