Toy "pull tab"-style feature on PCB
In "phone" connectors (also variously known as "headphone", "aux", or by their size (3.5 mm or 1/4") or contact configuration (TS or TRS)), many jacks have switch contacts that break a circuit when a plug is inserted. The simplest such configuration is one where the tip contact touches a switch contact unless a plug is inserted.

If you wire the switch contact to the sleeve contact of a TS (tip-sleeve, two conductor), then you have a circuit which is broken by inserting a plug, and the two contacts of the plug are the ends of the break point, exactly what you want.

You might choose to, instead of using TS plugs and jacks, use TRS, and carry the signal on the tip and ring. This has two advantages:


It provides a third conductor for common or shield purposes.
If there is a definite direction of signal flow, in which the tip is "to mod" and the ring is "from mod", then your usage is now identical to audio insert jacks, right down to the switching.




Picture for context: a 3.5 mm TRS jack with two switch contacts (top left for tip and middle just below the plug for ring).



(Photo taken by me. Licensed CC-BY-SA as per Stack Exchange terms. Please feel free to reuse elsewhere.)



Disadvantages of using these connectors:


When a plug is inserted or removed, temporary mis-connections/shorts occur.

If you use a TS style then the worst this can do is short the two contacts on the mod board (this is not a problem if it's unpowered). If you use the TRS "insert" style then this can also connect the mod board between the main board's output side and ground, which also shouldn't be too much of a problem. 

But, consider the electrical characteristics of your particular application. It may be easy enough to ensure that the outputs are current-limited and inputs are protected so that arbitrary misconnections don't damage anything — and this is a good idea for any exposed connectors, anyway!
The male connectors are usually on cable ends only. Board-mount male connectors exist (example I just searched for) but are not common parts with many varieties.
Depending on the design of the jack, sideways force may cause the sleeve connection to become intermittent. This is more commonly an issue on the 3.5 mm size than the 1/4" size, in my experience.

This could be reduced by choosing the right orientation of the jack versus any other mechanical interface features on the mod board. Or, it might not matter if the mod board has nothing else connected to it and isn't expected to be bumped in use.

If your mod can be a "dongle" on a cable, then this is not a problem and neither is the availability of connectors.