Here’s a very simple low-voltage, low-current latching continuity tester circuit. The probe voltage is only ~ 49 mV and probe current is 1/2 mA maximum, so it’s safe for testing almost any board without risk of damaging something.
R1/R2 and R3/R4 are voltage dividers; comparator U1 (an LM2903) turns on the beeper if the resistance between the probe leads is less than about 25 ohms.
I added C2 to make it latching; otherwise it’s very scratchy. The LM2903 is a smaller part than the original LM339, and I changed R4 from the original 10k ohms to 47k, to drop the continuity threshold from about 50 ohms to 25.
I used a TMB12A05 buzzer for LS1; these are available on eBay for about 20 cents each (quantity 10). Any small buzzer that buzzes by itself (with an internal oscillator) will probably work.
D1 and R5 just provide a power-on LED; you can omit them.
It’ll run on anything from about 3 to 15 volts (with the probe voltage about 1/100 of that); most any small-signal PNP transistor can be used for Q1.